Unknown Demand Distribution Research Articles

Nonparametric data-driven learning algorithms for multilocation inventory systems

We study a multilocation inventory system with unknown demand distribution using a nonparametric approach. The system consists of multiple distribution centers and customer locations, where products are shipped from the distribution centers to fulfill customer demands. We propose a novel algorithm, DMLI, for adaptive inventory management. Under specific conditions, we establish that the average expected T-period regret of DMLI converges to the optimal rate of O(1/T).

Data-Driven Algorithms for Two-Location Inventory Systems

In this paper, we consider a multiperiod, two-location inventory system with unknown demand distributions and perishable products. Products can be transshipped from the location with excess inventory to the other with excess demand to better fulfill customer demand. The demand distributions are assumed to follow a family of parametric distributions and can only be learned on the fly. To address the challenge, we propose a data-driven inventory management algorithm called DD2LI that achieves a good performance in terms of regret. This algorithm, DD2LI, employs maximum likelihood estimation to approximate the unknown parameter and determines the order quantity based on these estimations. In addition, we emphasize a key assumption that tightens regret bound. Finally, we test the effectiveness of our proposed algorithm by conducting numerical experiments for two scenarios.

A data-driven newsvendor model with unknown demand and supply distribution

A data-driven newsvendor model with unknown demand and supply distribution

Inventory Control and Learning for One-Warehouse Multistore System with Censored Demand

Efficient Learning Algorithms for Dynamic Inventory Allocation in Multiwarehouse Multistore Systems with Censored Demand Motivated by collaboration with a prominent fast-fashion retailer in Europe, the researchers focus their attention on the one-warehouse multistore (OWMS) inventory control problem, specifically addressing scenarios in which the demand distribution is unknown a priori. The OWMS problem revolves around a central warehouse that receives initial replenishments and subsequently distributes inventory to multiple stores within a finite time horizon. The objective lies in minimizing the total expected cost. To overcome the hurdles posed by the unknown demand distribution, the researchers propose a primal-dual algorithm that continuously learns from demand observations and dynamically adjusts inventory control decisions in real time. Thorough theoretical analysis and empirical evaluations highlight the promising performance of this approach, offering valuable insights for efficient inventory allocation within the ever-evolving retail industry.

An Automated Deep Reinforcement Learning Pipeline for Dynamic Pricing

Dynamic pricing problem is difficult due to the highly dynamic environment and unknown demand distributions. In this paper, we propose a Deep Reinforcement Learning (DRL) framework, which is a pipeline that automatically defines the DRL components for solving a Dynamic Pricing problem. The automated DRL pipeline is necessary because the DRL framework can be designed in numerous ways, and manually finding optimal configurations is tedious. The levels of automation make non-experts capable of using DRL for dynamic pricing. Our DRL pipeline contains three steps of DRL design, including MDP modeling, algorithm selection, and hyper-parameter optimization. It starts with transforming available information to state representation and defining reward function using a reward shaping approach. Then, the hyper-parameters are tuned using a novel hyper-parameters optimization method that integrates Bayesian Optimization and the selection operator of the Genetic algorithm. We employ our DRL pipeline on reserve price optimization problems in online advertising as a case study. We show that using the DRL configuration obtained by our DRL pipeline, a pricing policy is obtained whose revenue is significantly higher than the benchmark methods. The evaluation is performed by developing a simulation for the RTB environment that makes exploration possible for the RL agent.

Technical Note—Data-Driven Newsvendor Problem: Performance of the Sample Average Approximation

Sample Average Approximation in Data-Driven Newsvendor In the data-driven newsvendor problem, the manager makes sequential inventory decisions while learning the unknown demand distribution based on past demand samples. How does the widely used sample average approximation approach perform in this problem? In “Technical Note—Data-Driven Newsvendor Problem: Performance of the Sample Average Approximation,” Lin, Huh, Krishnan, and Uichanco analyze the performance of the sample average approximation as the time horizon grows, which turns out to be the best possible. The authors also examine how the local flatness of the demand distribution around the optimal order quantity affects the complexity of the problem. They show that the sample average approximation has the best achievable performance in terms of not only the time horizon, but also the local flatness of the demand distribution.

Optimality of Base Stock Policy under Unknown Demand Distributions: New Results and a New Method

This paper advances significantly the literature on the optimality of the base stock policy by generalizing the demand distribution and prior belief to be updated as demands, assumed to be continuous i.i.d. random variables, are observed over time. As the value function depends on the belief, the functional Bellman equation is infinite-dimensional. More importantly, we derive a functional equation for the derivative of the value function that provides a direct approach to obtain the optimal base stock. We characterize the way the base stock depends on the belief and implement it in two important cases. In the case of conjugate probabilities, the infinite-dimensional state reduces to a finite-dimensional sufficient statistic allowing us to solve numerically an example of Weibull demand. The second case considers the demand to come from one of two possible distributions, but we do not know which. This gives a functional equation in one hyperparameter expressing the ratio of the weights assigned to the two distributions. We develop an approximation scheme to solve it, obtain the formulas for the base stock, and show numerically that the procedure converges.

A Robust Data-Driven Approach for the Newsvendor Problem with Nonparametric Information

Problem definition: For the standard newsvendor problem with an unknown demand distribution, we develop an approach that uses data input to construct a distribution ambiguity set with the nonparametric characteristics of the true distribution, and we use it to make robust decisions. Academic/practical relevance: Empirical approach relies on historical data to estimate the true distribution. Although the estimated distribution converges to the true distribution, its performance with limited data is not guaranteed. Our approach generates robust decisions from a distribution ambiguity set that is constructed by data-driven estimators for nonparametric characteristics and includes the true distribution with the desired probability. It fits situations where data size is small. Methodology: We apply a robust optimization approach with nonparametric information. Results: Under a fixed method to partition the support of the demand, we construct a distribution ambiguity set, build a protection curve as a proxy for the worst-case distribution in the set, and use it to obtain a robust stocking quantity in closed form. Implementation-wise, we develop an adaptive method to continuously feed data to update partitions with a prespecified confidence level in their unbiasedness and adjust the protection curve to obtain robust decisions. We theoretically and experimentally compare the proposed approach with existing approaches. Managerial implications: Our nonparametric approach under adaptive partitioning guarantees that the realized average profit exceeds the worst-case expected profit with a high probability. Using real data sets from Kaggle.com, it can outperform existing approaches in yielding profit rate and stabilizing the generated profits, and the advantages are more prominent as the service ratio decreases. Nonparametric information is more valuable than parametric information in profit generation provided that the service requirement is not too high. Moreover, our proposed approach provides a means of combining nonparametric and parametric information in a robust optimization framework.

Partitioning cash flows to overcome retailer aversion to stocking new products

AbstractBecause decision makers tend to dislike ambiguity, the uncertainty surrounding new products can act as a barrier to retailer acceptance. We propose that by changing the structure of cash flows in the contract offered to a retailer (keeping net payments constant), a manufacturer can shift a retailer from making ambiguity‐averse to ambiguity‐neutral choices, thereby increasing the retailer's willingness to stock new products. In a series of studies, we demonstrate that contracts structured to provide positive cash flows after ambiguity has been resolved can increase the retailer's willingness to choose products with an unknown demand distribution. Thus, contracts with positive postdemand payments to the retailer have the potential to increase acceptance of new products.

Sampling-Based Approximation for Serial Multi-Echelon Inventory System

We consider inventory management of an infinite-horizon, serial system with unknown demand distribution. Each stage orders from its upstream stage and incurs inventory holding cost while the most downstream stage faces random demand and incurs inventory holding and demand backlogging cost. Lead times between stages are constants. The objective is to minimize the expected total discounted cost over the planning horizon. We apply the sample average approximation (SAA) method and study the performance of the resulting solution under the empirical distribution function constructed from a demand sample. We derive the sample size (i.e., distribution-free sample size) required to guarantee the performance of the SAA solution be close to the optimum with a given probability. This result is obtained by first deriving a decomposable and tight cost upper bound of the whole system that depends on the (given) echelon base-stock levels and then showing the cost difference between the SAA and optimal solutions can be measured by the distance between empirical and real demand distribution functions. Furthermore, when the one-period demand distribution function is absolutely continuous and has an increasing failure rate, we derive a smaller sample size (i.e., distribution-dependent sample size) that guarantees the same performance. A special case of this result for the newsvendor problem generalizes the existing result by Levi et al. (2015). In addition, both the distribution-free and the distribution-dependent sample sizes increase polynomially as the number of stages increases. Finally, our numerical study corroborates the theoretical results.}

The Nonstationary Newsvendor: Data-Driven Nonparametric Learning

We study a newsvendor problem with unknown demand distribution in a nonstationary demand environment over a multi-period time horizon. The demand in each period consists of a time-varying demand level and an additive random shock. Neither the demand level nor the random shock is separately observable. The amount of change in the demand level over the time horizon is measured by a cumulative variation metric. The problem has widespread applications, such as perishable inventory planning, staffing, and medical resource capacity planning in the wake of COVID-19. We design a nonparametric dynamic ordering policy, termed the moving window ordering policy, that tracks the shifts in the unknown demand level while accounting for the unobservable random demand shocks. To compute the order quantity in each period, this policy only needs the past demand observations, without any access to the underlying demand distribution. For a finite variation budget, we prove that our ordering policy is first-order optimal in the sense that its regret grows at the smallest possible rate. We also extend our analysis to the case of asymptotically large variation budgets. Through case studies based on real-life data, we show that our policy can save 20-80% of overage and underage costs, relative to policies widely used for perishable inventory replenishment and nurse staffing.

Simulation of the Worst Failure Probability of Sharing Platform under Unknown Demand Distribution

The sharing economy integrates the idle resources to match the fragmented market demand via the Internet platform. Different from the supply side, the demand is highly uncertain, in which, significant spikes and valleys exist. Therefore, it is assumed that only the statistical characteristics, instead of precise distributions, are known. The matching efficiency of supply and demand in sharing platform is studied. Firstly, a linear programming model under the moment constraints is proposed to describe the worst failure possibility in demand satisfaction. Then, the equivalent semi-infinite program is developed by Lagrange duality, and the analytical expression of the worst failure probability can be given. Finally, simulation study is implemented. This work not only riches the studies of moment-based bound, but also provides the managerial insights to the demand management of the sharing platform in the uncertain setting.

Network Revenue Management with Limited Switches: Known and Unknown Demand Distributions

Our work is motivated by a common business constraint in online markets. While firms respect the advantages of dynamic pricing and price experimentation, they must limit the number of price changes (i.e., switches) to be within some budget due to various practical reasons. We study both the classical price-based network revenue management problem in the distributionally-unknown setup, and the bandits with knapsacks problem. In these problems, a decision-maker (without prior knowledge of the environment) has finite initial inventory of multiple resources to allocate over a finite time horizon. Beyond the classical resource constraints, we introduce an additional switching constraint to these problems, which restricts the total number of times that the decision-maker makes switches between actions to be within a fixed switching budget. For such problems, we show matching upper and lower bounds on the optimal regret, and propose computationally-efficient limited-switch algorithms that achieve the optimal regret. Our work reveals a surprising result: the optimal regret rate is completely characterized by a piecewise-constant function of the switching budget, which further depends on the number of resource constraints --- to the best of our knowledge, this is the first time the number of resources constraints is shown to play a fundamental role in determining the statistical complexity of online learning problems. We conduct computational experiments to examine the performance of our algorithms on a numerical setup that is widely used in the literature. Compared with benchmark algorithms from the literature, our proposed algorithms achieve promising performance with clear advantages on the number of incurred switches. Practically, firms can benefit from our study and improve their learning and decision-making performance when they simultaneously face resource and switching constraints.

Controlling risk and demand ambiguity in newsvendor models

We use distributionally robust optimization (DRO) to model a general class of newsvendor problems with unknown demand distribution. The goal is to find an order quantity that minimizes the worst-case expected cost among an ambiguity set of distributions. The ambiguity set consists of those distributions that are not far—in the sense of the total variation distance—from a nominal distribution. The maximum distance allowed in the ambiguity set (called level of robustness) places the DRO between the risk-neutral stochastic programming and robust optimization models. An important problem a decision maker faces is how to determine the level of robustness—or, equivalently, how to find an appropriate level of risk-aversion. We answer this question in two ways. Our first approach relates the level of robustness and risk to the regions of demand that are critical (in a precise sense we introduce) to the optimal cost. Our second approach establishes new quantitative relationships between the DRO model and the corresponding risk-neutral and classical robust optimization models. To achieve these goals, we first focus on a single-product setting and derive explicit formulas and properties of the optimal solution as a function of the level of robustness. Then, we demonstrate the practical and managerial relevance of our results by applying our findings to a healthcare problem to reserve operating room time for cardiovascular surgeries. Finally, we extend some of our results to the multi-product setting and illustrate them numerically.

Driving inventory system simulations with limited demand data: Insights from the newsvendor problem

ABSTRACTStochastic inventory system simulation is often the tool of choice by industry practitioners who struggle with the evaluation of the quality of proposed inventory targets using service levels. However, driving simulations with unknown input distribution parameters has its own challenges. In this paper, we focus on the newsvendor problem and quantify the amount of demand parameter uncertainty – the uncertainty around the unknown demand distribution parameters which are estimated from the limited historical demand data – in the confidence interval of the mean service level. We use this quantification to understand how the variance of the mean service level, due to the amount of the demand parameter uncertainty in the simulation output process, changes with the choice of Type-1 and Type-2 service-level criteria, the historical data length, the ratio of the unit shortage cost to the unit holding cost, and the distributional shape of the demand’s density function.

Network design in scarce data environment using moment-based distributionally robust optimization

We consider a network design problem (NDP) under random demand with unknown distribution for which only a small number of observations are known. We design arc capacities in the first stage and optimize single-commodity network flows after realizing the demand in the second stage. The objective is to minimize the total cost of allocating arc capacities, flowing commodities, and penalty for unmet demand. We formulate a distributionally robust NDP (DR-NDP) by constructing an ambiguity set of the unknown demand distribution based on marginal moment information, to minimize the worst-case total cost over all possible distributions. Approximating polynomials with piecewise-linear functions, we reformulate DR-NDP as a mixed-integer linear program optimized via a cutting-plane algorithm. We test diverse network instances to compare DR-NDP with a stochastic programming approach, a deterministic benchmark model, and a robust NDP formulation. Our results demonstrate adequate robustness of optimal DR-NDP solutions and how they perform under varying demand, modeling parameter, network, and cost settings. The results highlight potential niche uses of DR-NDP in data-scarce contexts.

Simulation of inventory systems with unknown input models: a data-driven approach

Stochastic simulation is a commonly used tool by practitioners for evaluating the performance of inventory policies. A typical inventory simulation starts with the determination of the best-fit input models (e.g. probability distribution function of the demand random variable) and then obtains a performance measure estimate under these input models. However, this sequential approach ignores the uncertainty around the input models, leading to inaccurate performance measures, especially when there is limited historical input data. In this paper, we take an alternative approach and propose a simulation replication algorithm that jointly estimates the input models and the performance measure, leading to a credible interval for the performance measure under input-model uncertainty. Our approach builds on a nonparametric Bayesian input model and frees the inventory manager from making any restrictive assumptions on the functional form of the input models. Focusing on a single-product inventory simulation, we show that the proposed method improves the estimation of the service levels when compared to the traditional practice of using the best-fit or the empirical distribution as the unknown demand distribution.

A closed-form solution for the distribution free continuous review integrated inventory model

In this paper, we develop an integrated inventory model in a single-vendor single-buyer supply chain under an unknown demand distribution at the buyer. It is assumed that each lot delivered to the buyer contains a random fraction of defective items, and lead time can be reduced at an extra crashing cost. We also consider that the unmet demand at the buyer during the stockout period is partially backordered. A model is formulated to minimize the total expected relevant costs of the system considering an exact expression of the service level constraint to ensure that a certain percentage of customer orders are filled by the buyer. Closed-form expressions are derived for the optimal order quantity and safety factor for given lead time and shipment frequency, and then an algorithm is proposed to find the global optimal solution. Finally, a numerical example is presented to illustrate the solution procedure and sensitivity analysis is carried out to analyze the proposed model.

Robust Decisions of Capital-Constrained Supply Chain with Demand Uncertainty

In this paper, we design a supply chain finance system composed of a capital-constrained retailer, a manufacturer and a commercial bank, with unknown demand distribution. Considering the retailer's capital-constraint degree, credit line and bankruptcy risk, we establish a Stackelberg game in which the manufacturer acts as the leader and the commercial bank as the sub-leader. Based on the profit-maximization objective, we solve the optimal order of the retailer, the optimal interest rate of the commercial bank and the optimal wholesale price of the manufacturer and the robust deviation for each participant based on minimize-maximum-regret. Finally, we conduct numerical examples to make a comparative analysis between these two different decisions and explore the impact of the demand uncertainties on optimal strategies and robust deviations. It concluded that the robust decision provides a conservative strategy for risk aversion and the region of demand intervals has great impact on robust deviations, rather than the optimal decisions.

Load-dependent production planning for an assemble-to-order system

Assemble-to-order (ATO) systems enable manufacturing firms to be cost-effective and at the same time responsive to changing customer orders. In this paper, we study an assemble-to-order (ATO) system, where raw materials are released into a capacitated production stage. Inventory of produced components is kept in stock ready for final assembly once specific orders for products are received. An important planning issue is to find an optimal trade-off between capacity utilisation and components inventory holding cost with the aim to meet uncertain demand of end products. In this paper, we formulate two integrated production/inventory planning models using clearing functions while considering demand uncertainty. In the first model, we assume that demand is normally distributed and propose a chance constrained formulation. In the second model, we consider the case of unknown demand distributions and use a robust optimisation approach. The proposed models are linear programmes that provide safe production plans...