Lost Sales Inventory Model Research Articles

UCB-Type Learning Algorithms with Kaplan–Meier Estimator for Lost-Sales Inventory Models with Lead Times

Efficient Learning Algorithms for the Best Capped Base-Stock Policy in Lost Sales Inventory Systems Periodic review, lost sales inventory systems with lead times are notoriously challenging to optimize. Recently, the capped base-stock policy, which places orders to bring the inventory position up to the order-up-to level subject to the order cap, has demonstrated exceptional performance. In the paper “UCB-Type Learning Algorithms with Kaplan–Meier Estimator for Lost Sales Inventory Models with Lead Times,” Lyu, Zhang, and Xin propose an upper confidence bound–type learning framework. This framework, which incorporates simulations with the Kaplan–Meier estimator, works with censored demand observations. It can be applied to determine the optimal capped base-stock policy with a tight regret with respect to the planning horizon and the optimal base-stock policy with a regret that matches the best existing result. Both theoretical analysis and extensive numerical experiments demonstrate the effectiveness of the proposed learning framework.

Asymptotic Optimality of Open-Loop Policies in Lost-Sales Inventory Models with Stochastic Lead Times

Asymptotic Optimality of Open-Loop Policies in Lost-Sales Inventory Models with Stochastic Lead Times

Technical Note—Average Cost Optimality in Partially Observable Lost-Sales Inventory Systems

In many real-life situations, the inventory record may not match the actual stock perfectly. This can happen due to distortion of inventory data, such as transaction errors, misplaced inventories, and spoilage. In these cases, because the decision maker only has incomplete information about the inventory levels, many well-known inventory policies are not even admissible, and our understanding of the optimal policies, even their existence, is very limited. In “Average Cost Optimality in Partially Observable Lost-Sales Inventory Systems,” Bai et al. consider the classical lost-sales inventory model, in which the inventory level is only observed when it becomes zero. They formulate the cost-minimization problem as a partially observable Markov decision process. By exploiting the vanishing discount factor approach, they provide a way to verify the existence of optimal policies under the average cost criterion. The key step in their analysis is the construction of a valid policy, which, in a certain sense, copies the actions of another policy for the process starting from another initial state.

UCB-Type Learning Algorithms for Lost-Sales Inventory Models with Lead Times

In this paper, we consider a classic periodic-review lost-sales inventory system with lead times, which is notoriously challenging to optimize with a wide range of real-world applications. We consider a joint learning and optimization problem in which the decision-maker does not know the demand distribution a priori and can only use past sales information (i.e., censored demand). Departing from existing learning algorithms on this learning problem (e.g., Huh et al. 2009a, Agrawal and Jia 2019, Zhang et al. 2020) that require the convexity property of the underlining system, we develop an Upper Confidence Bound (UCB)-type learning framework and show it can be applied to the learning of not only the optimal base-stock policy, but also the optimal capped base-stock policy in which the convexity property no longer holds. Compared with a classic multi-armed bandit problem, our problem has unique challenges due to the nature of the inventory system, because (1) each action has long-term impacts on future costs, and (2) the system state space is exponentially large in the lead time. Hence, our learning algorithms are not naive adoptions of the classic UCB algorithm: the design of the simulation and averaging steps is novel in our algorithms, and the confidence width in the UCB index is also different from the classic one. We prove the regrets of our learning algorithms are tight, up to a logarithmic term, in the planning horizon T. Our extensive numerical experiments suggest the proposed algorithms (almost) dominate existing learning algorithms. We also propose a practical way to select which learning algorithm to use with limited demand data.

Technical Note—Understanding the Performance of Capped Base-Stock Policies in Lost-Sales Inventory Models

Single-sourcing lost-sales inventory systems with lead times are notoriously difficult to optimize. In this paper, we propose a new family of capped base-stock policies and provide a new perspective on constructing a practical hybrid policy combining two well-known heuristics: base-stock and constant-order policies. Each capped base-stock policy is associated with two parameters: a base-stock level and an order cap. We prove that for any fixed order cap, the capped base-stock policy converges exponentially fast in the base-stock level to a constant-order policy, providing a theoretical foundation for a phenomenon by which a capped dual-index policy converges numerically to a tailored base-surge policy recently observed in other work in a different but related dual-sourcing inventory model. As a consequence, there exists a sequence of capped base-stock policies that are asymptotically optimal as the lead time grows. We also numerically demonstrate its superior performance in general (including small lead times) by comparing it with otherwell-known heuristics.

Asymptotic Optimality of Semi-Open-Loop Policies in Markov Decision Processes with Large Lead Times

We consider a generic Markov decision process (MDP) with two controls: one control taking effect immediately and the other control whose effect is delayed by a positive lead time. Computing the optimal policy of this MDP is difficult when the lead time is large. Interestingly, as the lead time grows, one would naturally expect that the effect of the delayed action only weakly depends on the current state, and intuitively decoupling the delayed action from the current state could provide good controls. The purpose of this paper is to substantiate this decoupling intuition for some MDPs by establishing asymptotic optimality of the semi-open-loop policies, which specify open-loop controls for the delayed action and closed-loop controls for the immediate action. Specifically, we show that for an MDP with a fast mixing property and uniformly bounded cost functions, certain periodic semi-open-loop policies are asymptotically optimal. For a classical lost-sales inventory model with divisible products, we provide an elementary proof of asymptotic optimality of constant-order policies. For the same model with indivisible products and integral order quantities, we prove that a special integral open-loop policy, referred to as bracket policy, is asymptotically optimal. Our approach relies on a natural lower bound, provided by the optimal semi-open-loop policies for a finite-horizon problem with a horizon length equal to the original model lead time. We show that as the horizon length becomes large, the long-run average cost incurred by some specific semi-open-loop policies becomes close to the lower bound.

Technical Note—Constant-Order Policies for Lost-Sales Inventory Models with Random Supply Functions: Asymptotics and Heuristic

Asymptotic analysis of constant-order policies for lost-sales inventory models with positive lead times and random supply functions.

A 1.79-approximation Algorithm for a Continuous Review Lost-Sales Inventory Model

Stochastic inventory systems with lead times are often challenging to optimize, including single-sourcing lost-sales and dual-sourcing inventory systems. Recent numerical results have suggested that capped policies demonstrate superior performance over existing heuristics. However, the superior performance lacks a theoretical foundation, and why such policies generally perform so well remains a major open question. In this paper, we provide a theoretical foundation for this phenomenon in two classical inventory models with lead times. First, in a continuous review lost-sales inventory model with lead times and Poisson demand, we prove that a so-called capped base-stock policy has a worst-case performance guarantee of 1.79, by conducting an asymptotic analysis under a large penalty cost and lead time. Second, we extend the analysis to a more complex continuous review dual-sourcing inventory model with general lead times and Poisson demand, and also prove that a so-called capped dual-index policy has a worst-case performance guarantee of 1.79 under large lead time and ordering cost differences. Our results provide a deeper understanding of the superior numerical performance of capped policies, and present a new approach to proving worst-case performance guarantees of simple policies in hard inventory problems.

Using imperfect advance demand information in lost-sales inventory systems with the option of returning inventory

ABSTRACTMotivated by real-life applications, we consider an inventory system where it is possible to collect information about the quantity and timing of future demand in advance. However, this Advance Demand Information (ADI) is imperfect as (i) it may turn out to be false; (ii) a time interval is provided for the demand occurrences rather than its exact time; and (iii) there are still customer demand occurrences for which ADI cannot be provided. To make best use of imperfect information and integrate it with inventory supply decisions, we allow for returning excess stock built up due to imperfections to the upstream supplier and we propose a lost-sales inventory model with a general representation of imperfect ADI. A partial characterization of the optimal ordering and return policy is provided. Through an extensive numerical study, we investigate the value of ADI and factors that affect that value. We show that using imperfect ADI can yield substantial savings, the amount of savings being sensitive to the quality of information; the benefit of the ADI increases considerably if the excess stock can be returned. We apply our model to a spare parts case. The value of imperfect ADI turns out to be significant.

Constant-Order Policies for Lost-Sales Inventory Models with Random Supply Functions: Asymptotics and Heuristic

We consider an infinite-horizon lost-sales inventory model where the supply takes positive lead times and is a random function of the order quantity (e.g., random yield/capacity). The optimal policy for this model is computationally intractable; and no heuristic has been proposed in the literature. In this paper, we focus on a simple class of constant-order policies (COP) that place the same order in every period, regardless of the system state. Under some assumptions on the random supply function, we prove that the best COP is asymptotically optimal with large lead times and the optimality gap converges to zero exponentially fast in the lead time. We also prove that, if the mean supply capacity is less than the mean demand, then the best COP is also asymptotically optimal with large penalty costs; otherwise, the long-run average cost of the best COP asymptotically increases at the rate of square-root of the penalty cost. Further, we construct a simple heuristic COP and show that it performs very close to the best COP. Finally, we provide a numerical study to derive further insights into the performance of the best COP.

Lead time uncertainty and supply chain coordination in lost sales inventory models

The effects of lead time uncertainty on optimal inventory policies in a two-echelon supply chain operating under continuous review inventory assumptions are examined. In this system, the buyer selects an inventory order quantity and reorder point to maximise its profits, while the seller selects an order quantity from its external supplier that is an integer multiple of the buyer's order quantity; unsatisfied demand is accounted for as lost sales and may also incur a penalty cost representing lost customer goodwill. Normally distributed demand per unit time is assumed and lead time can follow any discrete probability distribution, resulting in a mixture of Gaussians distribution for lead time demand. Examples where lead time is discrete and not well-approximated by a continuous distribution are examined, and use of the mixture of Gaussians approach versus a single normal distribution for lead time demand in these cases reduces supply chain costs by between 12% and 13%.

A Variant of L♮-Convexity and its Application to Inventory Models with Batch Ordering

Previous studies show that the concept of L♮-convexity is helpful in characterizing the optimal policy for some inventory models with positive leadtimes. Such examples include the lost-sales inventory model by Zipkin (2008). On the structure of lost-sales inventory models. Operations Research, 56(4) 937–944. and the inventory-pricing model by Pang et al. (2012). A note on the structure of joint inventory-pricing control with leadtimes. Operations Research, 60(3), 581–587. However, when taking batch ordering into account, L♮-convexity does not work anymore. In this paper, we extend L♮-convexity to a more general concept termed as Q-jump-L♮-convexity and apply it to batch ordering inventory models including a lost-sales inventory model and an inventory-pricing model with batch ordering and positive leadtimes. By utilizing this new concept, we can partially characterize the structure of the optimal policies for both the models. Moreover, we are able to evaluate the sensitivity of the optimal decisions with respect to system states. Our results can also be applied to the serial and the assembly inventory systems with lost-sales and batch ordering.

Asymptotic optimality of constant capacity allocation policies for dynamic resource planning

We consider a stochastic capacity problem of dynamically matching the supply of resources and the uncertain demand for resources over a planning horizon of T periods. If demand exceeds the resource capacity in any time period, then such excess demand is lost and a penalty is incurred for the amount of lost demand. The discrete-time model reflects the relative time granularity at which resource capacity decisions can be made, whereas a lead time L > 0 represents in the model the relative time-scale difference between when a resource capacity decision is made and when this capacity decision takes effect. Our optimal control problem then is to determine the best policy for making resource capacity allocation decisions in each period, based on the demand realized up to that period, with the goal of maximizing netbenefit in expectation over the planning horizon. The net-benefit structure of each period in our model consists of a per-unit penalty for insufficient resource capacity to serve uncertain demand and a per-unit cost for the amount of resource capacity allocated. Such resource capacity problems arise within many different computing/networking applications, including capacity provisioning for cloud computing environments where high-frequency demand, having considerable uncertainty, needs to be served by resource capacity allocations over planning horizons during which any decisions to change the resource capacity allocation requires relatively long lead times. This resource capacity allocation model can be mapped in a mathematically equivalent manner to lost-sales inventory models [6]. In particular, the optimal order policy of the corresponding stochastic inventory control problem can be directly mapped to the optimal resource capacity policy, where the capacity allocation for any period is equivalent to the (nonnegative) inventory at the end of the period. We therefore consider henceforth our stochastic capacity planning problem within the context of stochastic lost-sales inventory control problems. Lost-sales inventory models have a long and rich history in the research literature, starting with their introduction by Bellman et al. [1]. Certain properties of the optimal policy have been investigated by various researchers; see, e.g., [8]. With respect to computation of the optimal policy, the primary approach has been dynamic programming, combined with various heuristics to speed up computations [7]. However, since the state-space of any such dynamic program

Session details: Special issue on the workshop on MAthematical performance Modeling and Analysis (MAMA 2014)

The complexity of computer systems, networks and applications, as well as the advancements in computer technology, continue to grow at a rapid pace. Mathematical analysis, modeling and optimization have been playing, and continue to play, an important role in research studies to investigate fundamental issues and trade-offs at the core of performance problems in the design and implementation of complex computer systems, networks and applications. On June 20, 2014, the 16th Workshop on MAthematical performance Modeling and Analysis (MAMA 2014) was held in Austin TX, USA, sponsored by ACM SIGMETRICS, and held in conjunction with SIGMETRICS 2014. This workshop seeks to bring together researchers working on the mathematical, methodological and theoretical aspects of performance analysis, modeling and optimization. It is intended to provide a forum at SIGMETRICS conferences for talks on early research in the more mathematical areas of computer performance analysis. These talks tend to be based on very recent research results (including work in progress) or on new research results that will be otherwise submitted only to a journal (or recently have been submitted to a journal). Thus, part of the goal is to complement and supplement the SIGMETRICS Conference program with such talks without removing any theoretical contributions from the main technical program. Furthermore, we continue to experience the desired result of having abstracts from previous MAMA workshops appear as full papers in the main program of subsequent SIGMETRICS and related conferences. All submissions were reviewed by at least 4 members of the program committee, from which a total of 13 were selected for presentation at the MAMA 2014 workshop. This special issue of Performance Evaluation Review includes extended abstracts relating to these presentations (arranged in the order of their presentation), which cover a wide range of topics in the area of mathematical performance analysis, modeling and optimization. The study of Gelenbe examines the backlog of energy and of data packets in a sensor node that harvests energy, computing the properties of energy and data backlogs and discussing system stability. Meyfroyt derives asymptotic results for the coverage ratio under a specific class of spatial stochastic models (Cooperative Sequential Adsorption) and investigates the scalability of the Trickle communication protocol algorithm. The study of Tune and Roughan applies the principle of maximum entropy to develop fast traffic matrix synthesis models, with the future goal of developing realistic spatio-temporal traffic matrices. Bradonjić et al. compare and contrast the capacity, congestion and reliability requirements for alternative connectivity models of large-scale data centers relative to fat trees. The study of Rochman et al. considers the problem of resource placement in network applications, based on a largescale service faced with regionally distributed demands for various resources in cloud computing. Xie and Lui investigate the design and analysis of a rating system and a mechanism to encourage users to participate in crowdsourcing and to incentivize workers to develop high-quality solutions. The study of Asadi et al. formulates a general problem for the joint per-user mode selection, connection activation and resource scheduling of connections using both LTE and WiFi resources within the context of device-todevice communications. Zheng and Tan consider a nonconvex joint rate and power control optimization to achieve egalitarian fairness (max-min weighted fairness) in wireless networks, exploiting the nonlinear Perron-Frobenius theory and nonnegative matrix theory. The study of Goldberg et al. derives an asymptotically optimal control policy for a stochastic capacity problem of dynamically matching supply resources and uncertain demand, based on connections with lost-sales inventory models. Ghaderi et al. investigate a dynamic stochastic bin packing problem, analyzing the fluid limits of the system under an asymptotic best-fit algorithm and showing it asymptotically minimizes the number of servers used in steady state. The study of Tizghadam and Leon-Garcia examines the impact of overlaying or removing a subgraph on the Moore-Penrose inverse of the Laplacian matrix of an existing network topology and proposes an iterative method to find key performance measures. Miyazawa considers a two-node generalized Jackson network in a phase-type setting as a special case of a Markov-modulated twodimensional reflecting random walk and analyzes the tail asymptotics for this reflecting process. The study of Squillante et al. investigates improvement in scalability of search in networks through the use of multiple random walks, deriving bounds on the hitting time to a set of nodes and on various performance metrics.

A stochastic multiobjective multiconstraint inventory model under inflationary condition and different inspection scenarios

This article presents inspection scenarios for the multiobjective multiconstraint mixed backorder and lost sales inventory model with imperfect items in which the order quantity, reorder point, ordering cost, lead time, and backorder rate are decision variables. The objectives are minimizing expected annual cost and variance of shortages. The number of imperfect items is assumed to be a beta-binomial random variable. There are two inspection scenarios: the imperfect items observed during inspection and screenings are either all reworked or all discarded. In order to fit some real environment, this study assumes the maximum permissible storage space and available budget are limited. Backorder rate is considered as a function of expected shortages at the end of cycle. Stochastic inflationary conditions with a probability density function are also considered in the presented model. This study assumes that the purchasing cost is paid when an order arrives at the beginning of the cycle, and the ordering cost is paid at the time of the order placing. The aggregate demand follows a normal distribution function. Finally, a solution procedure is proposed in order to solve the discussed multiobjective model. In addition, a numerical example is presented to illustrate the multiobjective model and its solution procedure for different inspection scenarios, and a sensitivity analysis is conducted with respect to the important system parameters.

Lead Time Reduction in Budget and Storage Space Restricted Lot Size Reorder Point Inventory Models with Controllable Negative Exponential Backorder Rate

This study presents the mixed backorder and lost sales inventory models involving four variables; order quantity, lead time, safety factor (a discrete variable) and backorder rate. A controllable negative exponential backorder rate is considered in the proposed model. In the real market, as unsatisfied demands occur, the longer length of lead time is, the smaller proportion of backorder rate would be. Considering this reason, backorder rate is dependent on the length of lead time through the amount of shortages. The negative exponential lead time crashing cost is considered in this study. Today, the cost of land acquisition is high in most of the countries and one of the main concerns of inventory managers is to ensure that the maximum permissible storage space is enough when an order arrives. Hence, a random storage space constraint is considered, since, the inventory level is random when an order arrives. So, in this case, a chance-constrained programming technique is used to make it crisp. Moreover, another significant concern of inventory managers is how to control the maximum investment in the inventory. This study assumes the purchasing cost is paid at the time of order placing. Considering this assumption, a budget constraint is also added to the model in order to managing the maximum inventory investment. The lead time demand, first, follows a normal distribution and then, relaxes the distribution function assumption by only assuming the mean and variance of lead time demand are known and applies the minimax distribution free procedure to solve the problem. Furthermore, a numerical example is also given to illustrate the models and solution procedures.

Technical Note—On the Quasiconcavity of Lost-Sales Inventory Models with Fixed Costs

We show that under a set of conditions, both the maximal profit function and the objective function in several lost-sales inventory models with fixed costs are quasiconcave. Not only is the quasiconcavity property useful computationally, it also leads to a sharper characterization of the optimal policies. Neither the proof of the quasiconcavity property itself nor the proof of the optimal policies by using the property requires the machinery of K-concavity or any of its K-related extensions, and hence they are intuitively appealing.

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Modified base-stock policies for continuous-review, lost-sales inventory models with Poisson demand and a fixed lead time

Abstract This paper reconsiders the lost-sales inventory system studied by Hill (2007) . The commonly assumed policy to apply to the system is a pure base-stock policy (PBSP) for which the best base stock is easily found. Hill shows that his simple delay policy (SDP) and full delay policy (FDP) perform better. The SDP is a (s,d) policy where s is the base stock of the best PBSP and d is a common lower bound on the delay between the placement of successive replenishment orders. We show by simulation that the d value suggested by Johansen (2001) outperforms Hill's suggestion and that the performance often can be further improved by optimizing d. For the test bed investigated by Hill, we show that, for some parameter settings, an additional improvement is achieved when s and d are optimized simultaneously. The policy suggested by Johansen performs better than the FDP in all settings where the former policy reduces the average cost of the best PBSP by at least 1%.