The joint order batching and picker routing problem is an important problem for improving warehouse efficiency. The goal is to minimize the total distance travel of picking customer orders. It is NP-hard, indicating that exact solutions are intractable for large instances. Many solution methods provided in the current literature use local search to solve the problem sequentially, often including a complex searching procedure for the order batching problem (OBP) followed by a simple heuristic for the picker routing problem (PRP). In this paper, we propose three heuristics that jointly solves the OBP and the PRP: two of these are combinations of variable neighborhood search, simulated annealing, and tabu search, while the third one uses guided local search. We discuss how the search procedure of the proposed algorithms can be parallelized, and benchmark them against state-of-the-art heuristics using well-studied instances. The results show a reduction of 3.06% to 7.63% in the geometric mean of the total travel distance across 64 instances. Increasing the number of threads in parallelization leads to diminishing returns in reducing running time and has no statistical effect on travel distance. In contrast, increasing the chunk size results in a linear increase in running time for all proposed algorithms, and improves the travel distance obtained when the batch size is 75 items.

This paper expands on previous machine learning techniques applied to combinatorial optimisation problems, to approximately solve the capacitated vehicle routing problem (VRP). We leverage the versatility of graph neural networks (GNNs) and extend the application of graph convolutional neural networks, previously used for the Travelling Salesman Problem, to address the VRP. Our model employs a supervised learning technique, utilising solved instances from the OR-Tools solver for training. It learns to provide probabilistic representations of the VRP, generating final VRP tours via non-autoregressive decoding with beam search. This work shows that despite that reinforcement learning based autoregressive approaches have better performance, GNNs show great promise to solve complex optimisation problems, providing a valuable foundation for further refinement and study.

Quantifying the uncertainty of a financial portfolio is important for investors and regulatory agencies. Reporting such uncertainty accurately is challenging due to time-dependent market dynamics, non-linearities in the return and risk properties of a portfolio, and due to the unobserved nature of the market risk. We propose Bayesian Neural Network (BNN) models, namely Recurrent Neural Network (RNN) and Long Short Term Memory (LSTM) models, to estimate the time-varying return distribution of an asset portfolio. The proposed models estimate the density of returns and incorporate parameter uncertainty through Bayesian inference. The uncertainty and any financial risk metric of interest can directly be obtained from the estimated density. Furthermore, through the BNN input-output design, proposed BNNs incorporate potential non-linear effects of each asset in the portfolio on the obtained density estimates. The proposed method is applicable to assess the uncertainty of any portfolio where the portfolio weight optimization is separated from risk assessment. We analyze the risk of a daily, equally weighted portfolio of 29 ETFs and a risk-free asset for a long time span with differing market environments between 09/06/2005 and 10/09/2020. We study the effects of different inference methods on the obtained results. The proposed models improve portfolio risk estimates compared to the benchmark. The performances of the proposed models depend on BNN design and the inference method. RNN models lead to relatively more stable results compared to LSTMs. Furthermore, the results of models with a relatively higher number of parameters depend heavily on the estimation method.

Dynamic portfolio optimization is a crucial but complex task due to financial market dynamics and the difficulty of disentangling noise from substantial changes in stock prices. In most existing methods, portfolios are re-optimized, hence re-balanced, at pre-specified time periods, return properties of each asset are dynamically computed, and portfolio weights are optimized according to an objective function. We propose a novel algorithm for dynamic portfolio optimization with a two-step signaling mechanism for re-balancing the portfolio including the optimization of re-balancing points and portfolio weights. The first step signals portfolio re-balancing only if there is a substantial price change in one or more of the portfolio constituents. These substantial price changes are defined according to directional change (DC) methods. DC methods create an intrinsic time series for each asset according to whether or not the change in the asset price exceeds a threshold level, hence removing part of the noise in asset prices. The second signaling mechanism uses genetic algorithms (GA) to assess if re-balancing is indeed profitable at each point indicated by the first signaling mechanism. The genetic algorithm is set up such that it simultaneously optimizes the weights of the re-balanced portfolio. For GA, we input the asset price summaries retrieved from DC methods to ensure that the GA can learn from the relatively less noisy data compared to observed asset prices. We show that the GA fit function can be set up to include several conventional trading strategies. As a first step, we apply the proposed method to a portfolio of 30 assets including 29 Exchange Traded Funds (ETF) and one risk-free asset where daily prices are observed during the period between 2 January 2018 and 30 December 2021. Second, we apply the method to 100 individual stocks for the same time period. We compare the obtained portfolio results with benchmarks, such as the simple buy and hold strategy of the S&P 500 index, the naive$1/N$portfolio, and a minimum variance portfolio in terms of standard portfolio evaluation methods including the Sharpe ratio.

Order picking is one of the most relevant optimization problems in the context of warehouse optimization. Especially within an e-commerce environment, order picking activities depend largely on the partitioning of individual customer orders into groups of orders that will be picked within a single pick tour, i.e., the order batching problem. The goal of this paper is to examine the optimization choices of the greedy seed-based batching algorithm. This algorithm follows a constructive, myopic approach in which batches are created consecutively by adding orders to the partial batch. It is widely used in different forms, but the reasons for its performance and the quality of its solution have not yet been addressed in the literature. We present a simulation study to investigate the properties of the individual pick tours that result from applying the seed-based batching algorithm. More specifically, we assess the optimality of the batching algorithm's myopic choices by comparing the myopic picking cost of orders that were the second best choice with their actual picking costs. Furthermore, we use this solution as a starting point for the variable neighborhood search algorithm to compare the cost of the myopic solutions. The results show that the decisions related to the choices of seed order, the constructive criterion to select orders to batch, and the storage policies all play major roles in the performance of the greedy seed-based algorithm.

The incidence of inflammatory bowel disease (IBD) is rising worldwide. Preventing disease progression by tight monitoring of disease activity using non-invasive procedures is important to prevent disease progression. In literature composite scores, combining biomarkers and clinical activity scores, have been described which aim to detect disease activity. In this paper we analyze the components of these composite scores in a data-driven manner using probabilistic fuzzy system (PFS).In this paper, we define a specific PFS for the analysis of the relationship between the biomarker fecal calprotectin (FC), with biomarkers erythrocyte sedimentation rate (ESR) and C-reactive protein (CRP) and clinical activity scores for two disease phenotypes, namely Crohn’s disease and ulcerative colitis.We report the relations between these biomarkers and clinical activity scores using the proposed PFS, and show how the statistical properties of FC indicator relates to the remaining biomarkers and clinical activity scores. Furthermore, we related the findings on the clinical activity scores to the conventional thresholds used in the literature to define susceptibility of disease activity to support clinical interpretation of results. The results show that analyzing the recently available data using PFS lead to valuable information for detecting IBD disease activity where most conventional thresholds for disease indicators are in line with the data-based findings.

Directional change (DC) representations of stock and exchange rate prices have been proposed as a new method for describing and forecasting intra-day price movements. In the DC approach, a time series price curve is transformed into an intrinsic time curve which records price changes that exceed a threshold level and identify upwards and downwards price changes. These price changes are shown to be relevant for creating investment strategies. In this paper we propose the use of a multi-output Probabilistic Fuzzy System (PFS) to forecast upwards and downwards price changes in the intrinsic time curve. The use of PFS allows to model the stochastic uncertainty of the price changes, while maintaining a linguistic description of the system. The proposed model forecasts upwards and downward movements depending on current market conditions. We apply the proposed method to 5-minute intraday data and report the accuracy of the model in estimating and forecasting price changes. In addition, we illustrate how the uncertainty in these forecasts can potentially be used to define DC-based investment strategies. We report the risk-return features of these strategies and compare them with a conventional moving window strategy and an existing DC investment strategy.

Vitamin B12 deficiency is a common disorder with severe impacts on hematological and neurological disorders. Identifying vitamin B12 deficiency is not straightforward since blood vitamin B12 levels are not representative for actual vitamin B12 status in tissue. Instead, methylmalonic acid (MMA) levels in the plasma are used as indicators of vitamin B12 deficiency. MMA concentrations increase starting from the early course of vitamin B12 deficiency but they may also be high regardless of vitamin B12 deficiency due to renal failure (measured by eGFR). In this paper we propose the use of probabilistic fuzzy systems (PFS) to explore the relationship between MMA plasma levels with vitamin B12 and kidney function. We propose a PFS model for the analysis of overall MMA properties for all patients and also specific MMA properties for individual patients. We show that this PFS model leads to accurate MMA interval predictions. We further show that the proposed model can be used to assess a change in the eGFR level to a normal eGFR level, and its effect on the patient's MMA distribution.

Probabilistic fuzzy systems (PFS) are shown to be valuable methods for conditional density estimation that combine fuzziness or linguistic uncertainty and probabilistic uncertainty. Several PFS applications have shown the added value of the different reasoning mechanisms of PFS and gains from incorporating two types of uncertainty. The effects of parametrization and parameter estimation on the function or conditional density approximations of PFS have not been documented in the literature. This paper aims to fill this gap in the literature by analyzing the parameters of PFS in conditional density estimation and point forecast using synthetic and real data applications. We show that both in-sample and out-of-sample results depend on PFS parametrization and the results deteriorate when the probability parameters of PFS are not optimized correctly, since these parameters allow the system to be fine tuned.

Two new methods for the optimization of probabilistic fuzzy classifiers are proposed. Probabilistic fuzzy systems are specially attractive due to their explicit and simultaneous modelling of two kinds of uncertainty, namely vagueness in linguistic terms (fuzziness) and probabilistic uncertainty. The current method uses the maximization of the likelihood with the stochastic gradient descent, which not only converges to local minima but also does not guarantee the minimization of the misclassification error. The proposed methods address this specific problem by incorporating global search techniques. The first algorithm proposed is a genetic algorithm with simple crossover and mutation operations. The other is a first generation memetic algorithm which combines the genetic algorithm with the stochastic gradient descent. A total of five benchmarks were used to compare the three algorithms. The results show that the proposed methods have an average relative improvement of 2% and 6% for the accuracy with the genetic and memetic algorithms, respectively.

Probabilistic fuzzy system (PFS) combines a linguistic description of the system behaviour with statistical properties of data. In this paper, we propose a multi-covariate multi-output PFS for explaining and forecasting quarterly US inflation data, which shows different patterns over time such as inflation level and volatility changes. An application of a PFS to model inflation was not considered in the literature. An important aspect in inflation forecasting for macroeconomic policy makers and financial institutions is obtaining accurate forecasts for the complete inflation density together with point forecasts of inflation. We present the first PFS application where estimation and forecasting capability of PFS is assessed based on point and density forecasts. The proposed PFS model is used to forecast one, four and eight quarters ahead inflation levels and densities. Additional information provided by the different interpretations of the PFS model is used to analyse changing inflation patterns over time. The linguistic description of PFS is particularly important for the input variables which depend individuals judgement and perception. It is found that the proposed model provides accurate point and density forecasts for US inflation. In addition, changing patterns in inflation density are captured by the proposed model.

Probabilistic fuzzy systems (PFS), a model which combines a linguistic description of the system behaviour with statistical properties of data, have been successfully applied to one day ahead Value at Risk (VaR) estimation for the stock market returns data. In this work, we propose a multi-covariate multi-output PFS model which provides the conditional density forecasts of returns for one day ahead and one month ahead periods. Such a multi-output PFS model was not considered in the literature. Furthermore, this model allows to analyze seasonal patterns in returns. The proposed model is applied to daily S&P500 stock returns. It is found that the proposed model indicates seasonal patterns in short and longer horizons as well as conservative VaR in long term forecasts. The model is shown to perform well in VaR estimation according to the unconditional coverage and independence tests.

Linguistic summarization is a data mining and knowledge discovery approach to extract patterns and sum up large volume of data into simple sentences. There is a large research in generating linguistic summaries which can be used to better understand and communicate about patterns, evolution and long trends in numerical, time series or labelled data. The objective of this work is to develop a computational system capable of automatically generating linguistic descriptions of time series data of septic shock patients containing labelled data, not only of the whole series, but also on the differences between subsets of the data. This is of particular interest in septic shock, as the differences between patients are not well understood. For this purpose we propose a new type of differential summaries, based on a numerical criterion assessing the characteristics of the summary on each subset of interest. Furthermore, this paper proposes an extension of linguistic summaries to provide temporal and categorical contextualization. This is of particular interest in healthcare to detect differences related to a condition or illness as well as the effectiveness of the administered treatment.

We consider conditional density approximation by fuzzy systems. Fuzzy systems are typically used to approximate deterministic functions in which the stochastic uncertainty is ignored. We propose probabilistic fuzzy systems (PFSs), in which the probabilistic nature of uncertainty is taken into account. These systems take also fuzzy uncertainty into account by their fuzzy partitioning of input and output spaces. We discuss an additive reasoning scheme for PFSs that leads to the estimation of conditional probability densities and prove how such fuzzy systems compute the expected value of this conditional density function. We show that some of the most commonly used fuzzy systems can compute the same expected output value, and we derive how their parameters should be selected in order to achieve this goal. The additional information and process understanding provided by the different interpretations of the PFS models are illustrated using a real-world example

Value at Risk (VaR) has been successfully estimated using single covariate probabilistic fuzzy systems (PFS), a method which combines a linguistic description of the system behaviour with statistical properties of data. In this paper, we consider VaR estimation based on a PFS model for density forecast of a continuous response variable conditional on a high-dimensional set of covariates. The PFS model parameters are estimated by a novel two-step process. The performance of the proposed model is compared to the performance of a GARCH model for VaR estimation of the S&P 500 index. Furthermore, the additional information and process understanding provided by the different interpretations of the PFS models are illustrated. Our findings show that the validity of GARCH models are sometimes rejected, while those of PFS models of VaR are never rejected. Additionally, the PFS model captures both instant and periods of high volatility, and leads to less conservative models.

In the present work, we propose the application of probabilistic fuzzy systems (PFS) to model the prediction of mortality in septic shock patients. This technique is characterized by the combination of the linguistic description of the system with the statistical properties of data. Preliminary results for this particular clinical problem point that PFS models, besides performing as accurately as first order Takagi-Sugeno fuzzy models, also provide probability measures that provide additional clinical information upon which physicians can act on.

In this paper we build on previous work related to predicting the MSCI EURO index based on content analysis of ECB statements. Our focus is on reducing the number of features employed for prediction through feature selection. For this purpose we rely on two methodologies: (stepwise) linear regression and greedy forward feature subset selection. The original dataset consists of 13 features (General Inquirer content categories). Both methodologies provide an improvement in the overall accuracy of the model, while reducing the number of features employed. Through linear regression we achieve an accuracy of 67.58% on the testing set by relying on six features, while greedy forward selection enables an accuracy on the test set of 69.50% while relying on eight features.

Real-world databases often contain missing data and existing correction algorithms deliver varying performance. Also, most modeling techniques are not suitable to deal with them automatically. In this study we examine different approaches to predicting septic shock in the presence of missing data. Some preprocessing techniques for managing missing data include disregarding data, or replacing it with information that by design introduces bias. In this study, we show that predictive performance improves by employing a minimum pre-processing technique, the Zero-Order-Hold (ZOH) method, by applying a Fuzzy C-Means clustering technique based on the partial distance calculation strategy (FCM-PDS) and by computing the final classification regarding the samples from each patient. Performance improvements continue to occur where up to approximately 60% of the data is missing, though for higher percentage the classification performance still is statistically improved. We further validate this approach by making comparisons with previous studies.

Real word data sets often contain many missing elements. Most algorithms that automatically develop a rule-based model are not well suited to deal with incomplete data. The usual technique is to disregard the missing values or substitute them by a best guess estimate, which can bias the results. In this paper we propose a new method for estimating the parameters of a Takagi-Sugeno fuzzy model in the presence of incomplete data. We also propose an inference mechanism that can deal with the incomplete data. The presented method has the added advantage that it does not require imputation or iterative guess-estimate of the missing values. This methodology is applied to fuzzy modeling of a classification and regression problem. The performance of the obtained models are comparable with the results obtained when using a complete data set.

In this paper we investigate whether the MSCI EURO index can be predicted based on the content of European Central Bank (ECB) statements. We propose a new model to retrieve information from free text and transform it into a quantitative output. For this purpose, we first identify all adjectives in an ECB statement by using the Stanford Part-of-Speech Tagger and feed these to the General Inquirer (GI) content analysis tool. From GI we obtain a matrix that provides for each document and for each content category the percentage of words in the document that fall under each category. After normalizing the data, we develop a Takagi-Sugeno (TS) fuzzy model using fuzzy c-means clustering. The TS fuzzy system is used to model the levels of the MSCI EURO index. To determine the performance of the model, we focus on the accuracy of predicting upward or downward movement in the index, and obtain, on average, an accuracy of 66%, that corresponds to an improvement of 16% over a random classifier.

This paper proposes extracting fuzzy rules from data using fuzzy possibilistic c-means and possibilistic fuzzy c-means algorithms, which provide more than one partition information: the typicality matrix and the membership matrix. Usually to extract fuzzy rules from data only one of the partition matrix is used, resulting in one rule per cluster. In our work we extract rules from both the membership partition matrix and the typicality matrix, resulting in deriving multiple rules for each cluster. These methods are applied to fuzzy modeling of four different classification problems: Iris, Wine, Wisconsin breast cancer and Altman data sets. The performance of the obtained models is compared and we consider the added value of the proposed approach in fuzzy modeling.

Value at risk (VaR) is a measure for senior management that summarises the financial risk a company faces into one single number. In this paper, we consider the use of fuzzy histograms for quantifying the value-at-risk of a portfolio. It is shown that the use of fuzzy histograms provides a good method of value-at-risk estimation for a portfolio of stocks. The conditional parameters of the model are obtained through minimisation of a test statistic for a VaR back testing method. Evolutionary optimisation is used for this purpose. It is found that statistical back testing always accepts fuzzy histogram models, while the popular GARCH models may be rejected.

The identification of fuzzy models for classification is a very complex task. Often, real world databases have a large number of features and the most relevant ones must be chosen. Therefore, it is necessary to select carefully the variables that are relevant for the feature class. A new automatic feature selection for classification problems is proposed in this paper, to construct compact fuzzy classification models. Several clustering algorithms are used and compared in terms of computational efficiency and accuracy in classification problems. The proposed algorithm was tested in well-known data sets: iris plant, wine, hepatitis, breast cancer and in a difficult real-world problem: the prediction of bankruptcy. The experiments show the advantages of the proposed method for selecting the proper features for classification.