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Governments have been challenged to provide timely medical care to face the COVID-19 pandemic. Under this pandemic, the demand for pharmaceutical products has changed significantly. Some of these products are in high demand, while, for others, their demand falls sharply. These changes in the random demand patterns are connected with changes in the skewness (asymmetry) and kurtosis of their data distribution. Such changes are critical to determining optimal lots and inventory costs. The lot-size model helps to make decisions based on probabilistic demand when calculating the optimal costs of supply using two-stage stochastic programming. The objective of this study is to evaluate how the skewness and kurtosis of the distribution of demand data, collected through sensors, affect the modeling of inventories of hospital pharmacy products helpful to treat COVID-19. The use of stochastic programming allows us to obtain results under demand uncertainty that are closer to reality. We carry out a simulation study to evaluate the performance of our methodology under different demand scenarios with diverse degrees of skewness and kurtosis. A case study in the field of hospital pharmacy with sensor-related COVID-19 data is also provided. An algorithm that permits us to use sensors when submitting requests for supplying pharmaceutical products in the hospital treatment of COVID-19 is designed. We show that the coefficients of skewness and kurtosis impact the total costs of inventory that involve order, purchase, holding, and shortage. We conclude that the asymmetry and kurtosis of the demand statistical distribution do not seem to affect the first-stage lot-size decisions. However, demand patterns with high positive skewness are related to significant increases in expected inventories on hand and shortage, increasing the costs of second-stage decisions. Thus, demand distributions that are highly asymmetrical to the right and leptokurtic favor high total costs in probabilistic lot-size systems.

We propose a methodology based on partial least squares (PLS) regression models using the beta distribution, which is useful for describing data measured between zero and one. The beta PLS model parameters are estimated with the maximum likelihood method, whereas a randomized quantile residual and the generalized Cook and Mahalanobis distances are considered as diagnostic methods. A simulation study is provided for evaluating the performance of these diagnostic methods. We illustrate the methodology with real-world mining data. The results obtained in this study based on the beta PLS model and its diagnostics may be of interest for the mining industry. . 305 : 306HUERTA ET AL. for more details on the beta distribution. Bertrand et al 6 proposed an extension of the normal PLS regression using the beta distribution and considering PLS-GLM, which we name beta PLS regression in short. To optimally find the number of PLS components, in normal PLS regression or PLS-GLM, the prediction error sum of squared (PRESS) method is often used, which is a cross-validation measure; see Li et al. 12 Other methods used for finding the number of PLS components in PLS-GLM focus on information criteria, which are based on ML methods; see Bastien et al. 9 Diagnostic analysis is a necessary aspect to be considered in all statistical modeling, once the estimation of parameters has been performed. This analysis is conducted to assess the suitability of the distributional assumption and the sensitivity and stability of the parameter estimation. Diagnostics can be performed by residuals and global influence methods, such as the generalized Cook and Mahalanobis distances. Residuals allow us to detect the distributional assumption and identify atypical cases, which may affect the results from the statistical analysis of data. Global influence methods remove cases and evaluate their effect on the fitted model globally; see Cook and Weisberg 13 and Chatterjee and Hadi. 14 For the use of residuals and other diagnostic methods in nonnormal models, including beta regression, see, for example, previous studies. [15][16][17][18][19][20][21][22][23] Kaolinite is a clay mineral, which is part of the group of industrial silicate minerals, collected in mining; see Orbovic and Huang. 24 Kaolinite is present in rock samples obtained by batch, and it is often measured by near-infrared (NIR) spectrometry with wavelengths between 350 and 2500 nm with 200 scans per sample in 20 seconds; see Burns and Ciurczak. 25 Data related to kaolinite are often measured between zero and one as a proportion. Then, these data can be adequately described by the beta distribution. However, proportion of kaolinite is explained by spectral variables, which usually correspond to a high number of covariates many of them correlated. Thus, a beta PLS regression should be used to model the proportion of kaolinite in function of spectral variables instead of a normal PLS regression.The main objective of this paper is to propose a methodology based on beta PLS regression to predict...

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