Optimal Base-stock Policy Research Articles

Asymptotic optimality of base-stock policies for lost-sales inventory systems with stochastic lead times

We consider the lost-sales inventory systems with stochastic lead times and establish the asymptotic optimality of base-stock policies for such systems. Specifically, we prove that as the per-unit lost-sales penalty cost becomes large compared to the per-unit holding cost, the ratio of the optimal base-stock policy's cost to the optimal cost converges to one. Our paper provides a theoretical guarantee of the widely adopted base-stock policies in lost-sales inventory systems with stochastic lead times for the first time.

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.

Inventory control with modulated demand and a partially observed modulation process

We consider a periodic review inventory control problem having an underlying modulation process that affects demand and that is partially observed by the uncensored demand process and a novel additional observation data (AOD) process. We present an attainability condition, AC, that guarantees the existence of an optimal myopic base stock policy if the reorder cost $$K=0$$ and the existence of an optimal (s, S) policy if $$K>0$$ , where both policies depend on the belief function of the modulation process. Assuming AC holds, we show that (i) when $$K=0$$ , the value of the optimal base stock level is constant within regions of the belief space and that each region can be described by two linear inequalities and (ii) when $$K>0$$ , the values of s and S and upper and lower bounds on these values are constant within regions of the belief space and that these regions can be described by a finite set of linear inequalities. A heuristic and bounds for the $$K=0$$ case are presented when AC does not hold. Special cases of this inventory control problem include problems considered in the Markov-modulated demand and Bayesian updating literatures.

Condition-based critical level policy for spare parts inventory management

Condition-based maintenance is a proactive maintenance strategy where the condition of machinery is monitored. A machine is preventively maintained when it shows signs of degradation, in which case the necessary spare parts are provided from stock. However, machines may occasionally suffer from sudden shocks, or they may fail before showing any signs of deterioration. Having stock-outs in case of failures leads to costly downtimes. To avoid such downtimes, we propose a condition-based critical level policy. In this policy, preventive maintenance can be carried out such that spare parts can be consumed when the stock level is not below the critical level. In order to assess the potential impact of this policy, we develop a conceptual model for which we establish a simple numerical procedure that allows to select the optimal base-stock and critical level thresholds. We compare the operational cost under the optimal critical level policy to that under the optimal base-stock policy. The results indicate that the proposed critical level policy can indeed reduce the operational cost significantly as compared to the base-stock policy.

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.

Stochastic optimization in supply chain networks: averaging robust solutions

We propose a novel robust optimization approach to analyze and optimize the expected performance of supply chain networks. We model uncertainty in the demand at the sink nodes via polyhedral sets which are inspired from the limit laws of probability. We characterize the uncertainty sets by variability parameters which control the degree of conservatism of the model, and thus the level of probabilistic protection. At each level, and following the steps of the traditional robust optimization approach, we obtain worst case values which directly depend on the values of the variability parameters. We go beyond the traditional robust approach and treat the variability parameters as random variables. This allows us to devise a methodology to approximate and optimize the expected behavior via averaging the worst case values over the possible realizations of the variability parameters. Unlike stochastic analysis and optimization, our approach replaces the high-dimensional problem of evaluating expectations with a low-dimensional approximation that is inspired by probabilistic limit laws. We illustrate our approach by finding optimal base-stock and affine policies for fairly complex supply chain networks. Our computations suggest that our methodology (a) generates optimal base-stock levels that match the optimal solutions obtained via stochastic optimization within no more than 4 iterations, (b) yields optimal affine policies which often times exhibit better results compared to optimal base-stock policies, and (c) provides optimal policies that consistently outperform the solutions obtained via the traditional robust optimization approach.

Technical Note—Perishable Inventory Systems: Convexity Results for Base-Stock Policies and Learning Algorithms Under Censored Demand

We develop the first nonparametric learning algorithm for periodic-review perishable inventory systems. In contrast to the classical perishable inventory literature, we assume that the firm does not know the demand distribution a priori and makes replenishment decisions in each period based only on the past sales (censored demand) data. It is well known that even with complete information about the demand distribution a priori, the optimal policy for this problem does not possess a simple structure. Motivated by the studies in the literature showing that base-stock policies perform near optimal in these systems, we focus on finding the best base-stock policy. We first establish a convexity result, showing that the total holding, lost sales and outdating cost is convex in the base-stock level. Then, we develop a nonparametric learning algorithm that generates a sequence of order-up-to levels whose running average cost converges to the cost of the optimal base-stock policy. We establish a square-root convergence rate of the proposed algorithm, which is the best possible. Our algorithm and analyses require a novel method for computing a valid cycle subgradient and the construction of a bridging problem, which significantly departs from previous studies. The e-companion is available at https://doi.org/10.1287/opre.2018.1724

Optimizing stock levels for rental systems with a support warehouse and partial backordering

Various rental systems, such as public libraries and tool rental companies, have a support warehouse for storing rental products at low cost and carrying out expedited shipments in response to stock-outs at local warehouses. In case of stock-outs, rental systems commonly employ partial backordering, i.e., a limited number of demands is backordered while additional demand is lost. We include partial backordering in our models and demonstrate its relevance. We fully characterize costs and optimal base stock policies in cases with one support and one local warehouse and provide upper bounds for base stock levels in the general system. By enumerating base stock policies and evaluating exact costs numerically, we determine optimal base stock policies for small systems with up to six local warehouses. For larger systems we develop a greedy algorithm based on approximate cost evaluations. The resulting base stock policies have costs within 0.2% from optimality on average in experiments. Among other things, experiments indicate that existing methods with no backordering may lead to poor solutions with partial backordering and that partial backordering is an effective means to reduce lost demand even with limited partial backorder levels.

Closing the Gap: A Learning Algorithm for Lost-Sales Inventory Systems with Lead Times

We consider a periodic-review, single-product inventory system with lost sales and positive lead times under censored demand. In contrast to the classical inventory literature, we assume the firm does not know the demand distribution a priori and makes an adaptive inventory-ordering decision in each period based only on the past sales (censored demand) data. The standard performance measure is regret, which is the cost difference between a learning algorithm and the clairvoyant (full-information) benchmark. When the benchmark is chosen to be the (full-information) optimal base-stock policy, Huh et al. [Huh WT, Janakiraman G, Muckstadt JA, Rusmevichientong P (2009a) An adaptive algorithm for finding the optimal base-stock policy in lost sales inventory systems with censored demand. Math. Oper. Res. 34(2):397–416.] developed a nonparametric learning algorithm with a cubic-root convergence rate on regret. An important open question is whether there exists a nonparametric learning algorithm whose regret rate matches the theoretical lower bound of any learning algorithms. In this work, we provide an affirmative answer to this question. More precisely, we propose a new nonparametric algorithm termed the simulated cycle-update policy and establish a square-root convergence rate on regret, which is proven to be the lower bound of any learning algorithm. Our algorithm uses a random cycle-updating rule based on an auxiliary simulated system running in parallel and also involves two new concepts, namely the withheld on-hand inventory and the double-phase cycle gradient estimation. The techniques developed are effective for learning a stochastic system with complex system dynamics and lasting impact of decisions. This paper was accepted by Yinyu Ye, optimization.

Closing the Gap: A Learning Algorithm for the Lost-Sales Inventory System with Lead Times

We consider a periodic-review single-product inventory system with lost-sales and positive lead times under censored demand. In contrast to the classical inventory literature, we assume the firm does not know the demand distribution a priori, and makes adaptive inventory ordering decision in each period based only on the past sales (censored demand) data. The standard performance measure is regret, which is the cost difference between a feasible learning algorithm and the clairvoyant (full-information) benchmark. When the benchmark is chosen to be the (full-information) optimal base-stock policy, Huh et al. [Mathematics of Operations Research 34(2): 397-416 (2009)] developed a nonparametric learning algorithm with a cubic-root convergence rate on regret. An important open question is whether there exists a nonparametric learning algorithm whose regret rate matches the theoretical lower bound of any learning algorithms. In this work, we provide an affirmative answer to the above question. More precisely, we propose a new nonparametric algorithm termed the simulated cycle-update policy, and establish a square-root convergence rate on regret, which is proven to be the lower bound of any learning algorithms. Our algorithm uses a random cycle-updating rule based on an auxiliary simulated system running in parallel, and also involves two new concepts, namely, the withheld on-hand inventory and the double-phase cycle gradient estimation. The techniques developed are effective for learning a stochastic system with complex systems dynamics and lasting impact of decisions.

동적 계획법을 이용한 LNG 현물시장에서의 포트폴리오 구성방법

Among many energy resources, natural gas has recently received a remarkable amount of attention, particularly from the electrical generation industry. This is in part due to increasing shale gas production, providing an environment-friendly fossil fuel, and high risk of nuclear power. Because South Korea, the world`s second largest LNG importing nation after Japan, has no international natural gas pipelines and relies on imports in the form of LNG, the natural gas has been traditionally procured by long term LNG contracts at relatively high price. Thus, there is a need of developing an Asian LNG trading hub, where LNG can be traded at more competitive spot prices. In a natural gas spot market, the amount of natural gas to be bought should be carefully determined considering a limited storage capacity and future pricing dynamics. In this work, the problem to find the optimal amount of natural gas in a spot market is formulated as a Markov decision process (MDP) in risk neutral environment and the optimal base stock policy which depends on a stage and price is established. Taking into account price and demand uncertainties, the basestock target levels are simply approximated from dynamic programming. The simulation results show that the basestock policy can be one of effective ways for procurement of LNG in a spot market.

An optimal inventory management problem with reputation-dependent demand

The paper addresses a problem of inventory control in a two-echelon retailer-customer setting. The problem incorporates the reputation of the retailer as a key factor influencing its policy. The retailer’s reputation depends on the degree to which past demand was satisfied. In turn, the reputation impacts the distribution of future demand. The optimality conditions developed by means of a perturbation analysis technique enable an optimal base-stock policy, in which the current level of base-stock depends on the retailer’s current reputation.

Optimal base-stock policy of the assemble-to-order systems

In this work, an ordinal optimization-based evolution algorithm (OOEA) is proposed to solve a problem for a good enough target inventory level of the assemble-to-order (ATO) system. First, the ATO system is formulated as a combinatorial optimization problem with integer variables that possesses a huge solution space. Next, the genetic algorithm is used to select N excellent solutions from the solution space, where the fitness is evaluated with the radial basis function network. Finally, we proceed with the optimal computing budget allocation technique to search for a good enough solution. The proposed OOEA is applied to an ATO system comprising 10 items on 6 products. The solution quality is demonstrated by comparing with those obtained by two competing methods. The good enough target inventory level obtained by the OOEA is promising in the aspects of solution quality and computational efficiency.

A belief-rule-based inventory control method under nonstationary and uncertain demand

This paper is devoted to investigating inventory control problems under nonstationary and uncertain demand. A belief-rule-based inventory control (BRB-IC) method is developed, which can be applied in situations where demand and demand-forecast-error (DFE) do not follow certain stochastic distribution and forecasting demand is given in single-point or interval styles. The method can assist decision-making through a belief-rule structure that can be constructed, initialized and adjusted using both manager’s knowledge and operational data. An extended optimal base stock (EOBS) policy is proved for initializing the belief-rule-base (BRB), and a BRB-IC inference approach with interval inputs is proposed. A numerical example and a case study are examined to demonstrate potential applications of the BRB-IC method. These studies show that the belief-rule-based expert system is flexible and valid for inventory control. The case study also shows that the BRB-IC method can compensate DFE by training BRB using historical demand data for generating reliable ordering policy.

Inventory systems with stochastic demand and supply: Properties and approximations

We model a retailer whose supplier is subject to complete supply disruptions. We combine discrete-event uncertainty (disruptions) and continuous sources of uncertainty (stochastic demand or supply yield), which have different impacts on optimal inventory settings. This prevents optimal solutions from being found in closed form. We develop a closed-form approximate solution by focusing on a single stochastic period of demand or yield. We show how the familiar newsboy fractile is a critical trade-off in these systems, since the optimal base-stock policies balance inventory holding costs with the risk of shortage costs generated by a disruption.

An Adaptive Algorithm for Finding the Optimal Base-Stock Policy in Lost Sales Inventory Systems with Censored Demand

We consider a periodic-review, single-location, single-product inventory system with lost sales and positive replenishment lead times. It is well known that the optimal policy does not possess a simple structure. Motivated by recent results showing that base-stock policies perform well in these systems, we study the problem of finding the best base-stock policy in such a system. In contrast to the classical inventory literature, we assume that the manager does not know the demand distribution a priori but must make the replenishment decision in each period based only on the past sales (censored demand) data. We develop a nonparametric adaptive algorithm that generates a sequence of order-up-to levels whose running average of the inventory holding and lost sales penalty cost converges to the cost of the optimal base-stock policy, and we establish the cubic-root convergence rate of the algorithm. Our analysis is based on recent advances in stochastic online convex optimization and on the uniform ergodicity of Markov chains associated with bases-stock policies.

Stock Positioning and Performance Estimation for Distribution Systems with Service Constraints

The problem of minimizing average inventory costs in a distribution system consisting of one warehouse and multiple retailers subject to fill rate service constraints is studied. An effective algorithm to evaluate the optimal base-stock policy and cost under local control is provided. Properties of the optimal base-stock levels at each location are established. In addition, heuristics and closed-form approximations that yield insights into the drivers of system performance are also provided. An extensive numerical study to gain insight into the optimal solution, stock positioning and when to use the heuristics and approximations is presented. An example of how logistics postponement and fill-rate constraints affect the distribution system’s performance is quantified. A discussion on how to study and allocate stocks across an arborescence system with more than two echelons is presented.

Stock positioning and performance estimation for distribution systems with service constraints

The problem of minimizing average inventory costs in a distribution system consisting of one warehouse and multiple retailers subject to fill rate service constraints is studied. An effective algorithm to evaluate the optimal base-stock policy and cost under local control is provided. Properties of the optimal base-stock levels at each location are established. In addition, heuristics and closed-form approximations that yield insights into the drivers of system performance are also provided. An extensive numerical study to gain insight into the optimal solution, stock positioning and when to use the heuristics and approximations is presented. An example of how logistics postponement and fill-rate constraints affect the distribution system's performance is quantified. A discussion on how to study and allocate stocks across an arborescence system with more than two echelons is presented.

Inventory Systems with Stochastic Demand and Supply: Properties and Approximations

We model a retailer whose supplier is subject to complete supply disruptions. We combine discrete-event uncertainty (disruptions) and continuous sources of uncertainty (stochastic demand or supply yield), which have different impacts on optimal inventory settings. This prevents optimal solutions from being found in closed form. We develop a closed-form approximate solution by focusing on a single stochastic period of demand or yield. We show how the familiar newsboy fractile is a critical trade-off in these systems, since the optimal base-stock policies balance inventory holding costs with the risk of shortage costs generated by a disruption.

Optimal base-stock policy for the inventory system with periodic review, backorders and sequential lead times

We extend well-known formulae for the optimal base stock of the inventory system with continuous review and constant lead time to the case with periodic review and stochastic, sequential lead times. Our extension uses the notion of the 'extended lead time'. The derived performance measures are exact for Poisson demands.