Evaluating alternative frequentist inferential approaches for optimal order quantities in the newsvendor model under exponential demand

Three estimation policies for the optimal order quantity of the classical newsvendor model under exponential demand are evaluated in the current paper. According to the principle of the first estimation policy, the corresponding estimator is obtained replacing in the theoretical formula which gives the optimal order quantity the parameter of exponential distribution with its maximum likelihood estimator. The estimator of the second estimation policy is derived in such a way as to ensure that the requested critical fractile is attained. For the third estimation policy, the corresponding estimator is obtained maximizing the a-priori expected profit with respect to a constant which has been included into the form of the estimator. Three statistical measures have been chosen to perform the evaluation. The actual critical fractile attained by each estimator, the mean square error, and the range of deviation of estimates from the optimal order quantity, when the probability to take such a range is the same for the three estimation policies. The behavior of the three statistical measures is explored under different combinations of sample sizes and critical fractiles. With small sample sizes, no estimation policy predominates over the others. The estimator which attains the closest actual critical fractile to the requested one, this estimator has the largest mean square and the largest range of deviation of estimates from the optimal order quantity. On the contrary, with samples over 40 observations, the choice is restricted among the estimators of the first and third estimation policy. To facilitate this choice, at different sample sizes, we offer the required values of the critical fractile which determine which estimation policy eventually should be applied

Evaluating alternative frequentist inferential approaches for optimal order quantities in the newsvendor model under exponential demand

Three estimation policies for the optimal order quantity of the classical newsvendor model under exponential demand are evaluated in the current paper. According to the principle of the first estimation policy, the corresponding estimator is obtained replacing in the theoretical formula which gives the optimal order quantity the parameter of exponential distribution with its maximum likelihood estimator. The estimator of the second estimation policy is derived in such a way as to ensure that the requested critical fractile is attained. For the third estimation policy, the corresponding estimator is obtained maximizing the a-priori expected profit with respect to a constant which has been included into the form of the estimator. Three statistical measures have been chosen to perform the evaluation. The actual critical fractile attained by each estimator, the mean square error, and the range of deviation of estimates from the optimal order quantity, when the probability to take such a range is the same for the three estimation policies. The behavior of the three statistical measures is explored under different combinations of sample sizes and critical fractiles. With small sample sizes, no estimation policy predominates over the others. The estimator which attains the closest actual critical fractile to the requested one, this estimator has the largest mean square and the largest range of deviation of estimates from the optimal order quantity. On the contrary, with samples over 40 observations, the choice is restricted among the estimators of the first and third estimation policy. To facilitate this choice, at different sample sizes, we offer the required values of the critical fractile which determine which estimation policy eventually should be applied

Unbiased estimation of maximum expected profits in the Newsvendor Model: a case study analysis

In the current paper we study a real life inventory problem whose operating conditions match to the principles of the classical newsvendor model. Applying appropriate tests to the available sample of historical demand data, we get the sufficient statistical evidences to support that daily demand is stationary, uncorrelated, and normally distributed. Given that at the start of each day, the selling price, the purchasing cost per unit, and the salvage value are known, and do not change through the whole period under investigation, we derive exact and asymptotic prediction intervals for the daily maximum expected profit. To evaluate their performance, we derive the analytic form of three accuracy information metrics. The first metric measures the deviation of the estimated probability of no stock-outs during the day from the critical fractile. The other two metrics relate the validity and precision of the two types of prediction interval to the variability of estimates for the ordered quantity. Both theoretical and empirical analysis demonstrates the importance of implications of the loss of goodwill to the adopted inventory policy. Operating the system at the optimal situation, this intangible cost element determines the probability of no stock-outs during the day, and assesses the precision of prediction intervals. The rising of the loss of goodwill leads to smaller estimates for the daily maximum expected profit and to wider prediction intervals. Finally, in the setting of the real life newsvendor problem, we recommend the asymptotic prediction interval since with samples over 25 observations this type of interval has higher precision and probability to include the daily maximum expected profit almost equal to the nominal confidence level

Forecasting the optimal order quantity in the newsvendor model under a correlated demand

This paper considers the classical newsvendor model when, (a) demand is autocorrelated, (b) the parameters of the marginal distribution of demand are unknown, and (c) historical data for demand are available for a sample of successive periods. An estimator for the optimal order quantity is developed by replacing in the theoretical formula which gives this quantity the stationary mean and the stationary variance with their corresponding maximum likelihood estimators. The statistical properties of this estimator are explored and general expressions for prediction intervals for the optimal order quantity are derived in two cases: (a) when the sample consists of two observations, and (b) when the sample is considered as sufficiently large. Regarding the asymptotic prediction intervals, specifications of the general expression are obtained for the time-series models AR(1), MA(1), and ARMA(1,1). These intervals are estimated in finite samples using in their theoretical expressions, the sample mean, the sample variance, and estimates of the theoretical autocorrelation coefficients at lag one and lag two. To assess the impact of this estimation procedure on the optimal performance of the newsvendor model, four accuracy implication metrics are considered which are related to: (a) the mean square error of the estimator, (b) the accuracy and the validity of prediction intervals, and (c) the actual probability of running out of stock during the period when the optimal order quantity is estimated. For samples with more than two observations, these metrics are evaluated through simulations, and their values are presented to appropriately constructed tables. The general conclusion is that the accuracy and the validity of the estimation procedure for the optimal order quantity depends upon the critical fractile, the sample size, the autocorrelation level, and the convergence rate of the theoretical autocorrelation function to zero

Forecasting the optimal order quantity in the newsvendor model under a correlated demand

This paper considers the classical newsvendor model when, (a) demand is autocorrelated, (b) the parameters of the marginal distribution of demand are unknown, and (c) historical data for demand are available for a sample of successive periods. An estimator for the optimal order quantity is developed by replacing in the theoretical formula which gives this quantity the stationary mean and the stationary variance with their corresponding maximum likelihood estimators. The statistical properties of this estimator are explored and general expressions for prediction intervals for the optimal order quantity are derived in two cases: (a) when the sample consists of two observations, and (b) when the sample is considered as sufficiently large. Regarding the asymptotic prediction intervals, specifications of the general expression are obtained for the time-series models AR(1), MA(1), and ARMA(1,1). These intervals are estimated in finite samples using in their theoretical expressions, the sample mean, the sample variance, and estimates of the theoretical autocorrelation coefficients at lag one and lag two. To assess the impact of this estimation procedure on the optimal performance of the newsvendor model, four accuracy implication metrics are considered which are related to: (a) the mean square error of the estimator, (b) the accuracy and the validity of prediction intervals, and (c) the actual probability of running out of stock during the period when the optimal order quantity is estimated. For samples with more than two observations, these metrics are evaluated through simulations, and their values are presented to appropriately constructed tables. The general conclusion is that the accuracy and the validity of the estimation procedure for the optimal order quantity depends upon the critical fractile, the sample size, the autocorrelation level, and the convergence rate of the theoretical autocorrelation function to zero

Operational Decision Making under Uncertainty: Inferential, Sequential, and Adversarial Approaches

Modern security threats are characterized by a stochastic, dynamic, partially observable, and ambiguous operational environment. This dissertation addresses such complex security threats using operations research techniques for decision making under uncertainty in operations planning, analysis, and assessment. First, this research develops a new method for robust queue inference with partially observable, stochastic arrival and departure times, motivated by cybersecurity and terrorism applications. In the dynamic setting, this work develops a new variant of Markov decision processes and an algorithm for robust information collection in dynamic, partially observable and ambiguous environments, with an application to a cybersecurity detection problem. In the adversarial setting, this work presents a new application of counterfactual regret minimization and robust optimization to a multi-domain cyber and air defense problem in a partially observable environment

Evaluating alternative Frequentist inferential approaches for optimal order quantities in the newsvendor model under Exponential demand

Three estimation policies for the optimal order quantity of the classical newsvendor model when the demand is Exponentially distributed are evaluated in this paper. The evaluation is performed analytically for different combinations of sample sizes and values of the requested critical fractile. The statistical measures that have been chosen to perform the evaluation are (a) the actual critical fractile, namely, the actual probability for the estimated order quantities to meet the demand of the period, (b) the mean square error of the estimators for the optimal order quantity, and (c) the range of deviations of estimated order quantities from the optimal order quantity, provided that the probability of taking such a range is the same for the three estimation policies. For small and moderate sample sizes, no estimation policy predominates over the other two approaches, and the choice should be made on a subjective base according to the individual preferences of researchers or practitioners

Sequential data inference via matrix estimation : causal inference, cricket and retail

Thesis: Ph. D., Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2018.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (pages 185-193).This thesis proposes a unified framework to capture the temporal and longitudinal variation across multiple instances of sequential data. Examples of such data include sales of a product over a period of time across several retail locations; trajectories of scores across cricket games; and annual tobacco consumption across the United States over a period of decades. A key component of our work is the latent variable model (LVM) which views the sequential data as a matrix where the rows correspond to multiple sequences while the columns represent the sequential aspect. The goal is to utilize information in the data within the sequence and across different sequences to address two inferential questions: (a) imputation or "filling missing values" and "de-noising" observed values, and (b) forecasting or predicting "future" values, for a given sequence of data. Using this framework, we build upon the recent developments in "matrix estimation" to address the inferential goals in three different applications. First, a robust variant of the popular "synthetic control" method used in observational studies to draw causal statistical inferences. Second, a score trajectory forecasting algorithm for the game of cricket using historical data. This leads to an unbiased target resetting algorithm for shortened cricket games which is an improvement upon the biased incumbent approach (Duckworth-Lewis-Stern). Third, an algorithm which leads to a consistent estimator for the time- and location-varying demand of products using censored observations in the context of retail. As a final contribution, the algorithms presented are implemented and packaged as a scalable open-source library for the imputation and forecasting of sequential data with applications beyond those presented in this work.by Muhammad Jehangir Amjad.Ph. D