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Shannon’s entropy is usually defined separately for discrete, and for (absolutely) continuous random variables. However, many random variables encountered in practice have mixed (discrete‐continuous) distributions. A simple expression for the entropy of random variables with mixed (discrete‐continuous) distributions is derived in terms of the usual entropy definitions. In addition, the maximum entropy problem in the general setting of mixture distributions is discussed.

In real-world cases, it is common to encounter mixed discrete-continuous problems where some or all of the variables may take only discrete values. To solve these non-linear optimization problems, the use of finite element methods is very time-consuming. The purpose of this study is to investigate the efficiency of the proposed hybrid algorithms for the mixed discrete-continuous optimization and compare it with the performance of genetic algorithms (GAs).

In this paper, the enhanced multipoint approximation method (MAM) is used to reduce the original nonlinear optimization problem to a sequence of approximations. Then, the sequential quadratic programing technique is applied to find the continuous solution. Following that, the implementation of discrete capability into the MAM is developed to solve the mixed discrete-continuous optimization problems.

The efficiency and rate of convergence of the developed hybrid algorithms outperforming GA are examined by six detailed case studies in the ten-bar planar truss problem, and the superiority of the Hooke–Jeeves assisted MAM algorithm over the other two hybrid algorithms and GAs is concluded.

The authors propose three efficient hybrid algorithms, the rounding-off, the coordinate search and the Hooke–Jeeves search-assisted MAMs, to solve nonlinear mixed discrete-continuous optimization problems. Implementations include the development of new procedures for sampling discrete points, the modification of the trust region adaptation strategy and strategies for solving mix optimization problems. To improve the efficiency and effectiveness of metamodel construction, regressors f defined in this paper can have the form in common with the empirical formulation of the problems in many engineering subjects.

Many consumer choice situations are characterized by the simultaneous demand for multiple alternatives that are imperfect substitutes for one another. A simple and parsimonious multiple discrete-continuous extreme value (MDCEV) econometric approach to handle such multiple discreteness was formulated by Bhat (2005) within the broader Kuhn–Tucker (KT) multiple discrete-continuous economic consumer demand model of Wales and Woodland (1983). In this chapter, the focus is on presenting the basic MDCEV model structure, discussing its estimation and use in prediction, formulating extensions of the basic MDCEV structure, and presenting applications of the model. The paper examines several issues associated with the MDCEV model and other extant KT multiple discrete-continuous models. Specifically, the paper discusses the utility function form that enables clarity in the role of each parameter in the utility specification, presents identification considerations associated with both the utility functional form as well as the stochastic nature of the utility specification, extends the MDCEV model to the case of price variation across goods and to general error covariance structures, discusses the relationship between earlier KT-based multiple discrete-continuous models, and illustrates the many technical nuances and identification considerations of the multiple discrete-continuous model structure. Finally, we discuss the many applications of MDCEV model and its extensions in various fields.

The link between the magnitude of a bandwidth and the relevance of the corresponding covariate in a regression has recently garnered theoretical attention. Theory suggests that variables included erroneously in a regression will be automatically removed when bandwidths are selected via cross-validation procedure. However, the connections between the bandwidths of the variables that are smoothed away and the insights from these same variables when properly tested for statistical significance have not been previously studied. This paper proposes a variety of simulation exercises to examine the relative performance of both cross-validated bandwidths and individual and joint tests of significance. We focus on settings where the hypothesis of interest may focus on a single data type (e.g., continuous only) or a mix of discrete and continuous variables. Moreover, we propose an extension of a well-known kernel smoothing significance test to handle mixed data types. Our results suggest that individual tests of significance and variable-specific bandwidths are very close in performance, but joint tests and joint bandwidth recognition produce substantially different results. This underscores the importance of testing for joint significance when one is trying to arrive at the final nonparametric model of interest.

The paper deals with a problem of control of a mixed, discrete‐continuous dynamic process. The set of input data for such a process is finite. Control of such a process is very difficult due to the non‐polynomial computational complexity of the used algorithm. The paper suggests a formulation of the optimization problem as well as a method for its solution. The described method is applied to an industrial process, and particularly to the system conditioned by time for which the set of control operations changes according to the realization of the production process.

This paper aims to identify the key drivers of occupancy rates in peer-to-peer accommodation.

The applied methodology fits the specific characteristics of this market segment: the peculiar distribution of the occupancy rate (a ratio characterised by a large share of zeros) requires the adoption of a mixed discrete-continuous model; the insidious issue of price endogeneity is dealt with a control function approach; the econometric specification takes into account the monopolistic competition, the relevant market regime in the hospitality industry. The model is tested on Airbnb listings in the Balearic Islands (Spain).

The occupancy rate of peer-to-peer properties in the Balearic Islands strongly depends on their geographical location and online reputation. There is a qualitative difference between two groups: listings with positive occupancy rates, which demand tends to be inelastic, and listings with zero occupancy. The authors found that the price is a not a statistically significant determinant of the latter group membership.

This paper applies a zero-inflated beta model, never used in previous analyses of occupancy rates, to provide a benchmark for future studies. This procedure allows the estimation of unbiased marginal effects. It, thus, offers important technical and managerial implications, as a wrong understanding of how occupancy depends on price would deliver ineffective managerial decisions. This paper highlights the importance of methodological choices, as coefficients are highly sensitive to misspecifications of the model.

The purpose of this paper is to explore the problem of non-convex labor supply decision in an economy with both discrete and continuous labor decisions. In contrast to the setup in McGrattan et al. (1997), here each household faces an indivisible labor supply choice in the market sector, while it can choose to work any number of hours in the non-market sector.

The authors show how lotteries as in Rogerson (1988) can again be used to convexify consumption sets, and aggregation over individual preferences.

With a mix of discrete and continuous labor supply decisions, disutility of non-market work becomes separable from market work, and the elasticity of the latter increases from unity to infinity.

As a possible venue for future research, the authors plan to feed the derived aggregate utility function above in a sophisticated real-business-cycle model to investigate the effect of those preferences for the transmission of technology and fiscal shocks.

This is a novel and interesting result in the aggregation literature.

This paper first reviews factor reconstructability analysis (RA) modelling completely by data. Gives a description of levelled variable factor reconstruction analysis model generation process and its further extension to forecasting, evaluation and optimisation. It introduces RA modelling with multi‐variety information and knowledge. From generation of mixed variable RA models to generation of models with both data and knowledge, the paper gives a related table and a figure on “Information flow of reconstructability analysis with multi‐variety information and knowledge” which included database, RA relational knowledge base, and interfaces for input of experts' knowledge. Finally, it gives some examples and prospects.

Value‐at‐Risk (VaR) has become a mainstream risk management technique employed by a large proportion of financial institutions. There exists a substantial amount of research dealing with this task, most commonly referred to as VaR backtesting. A new generation of “self‐learning” VaR models (Conditional Autoregressive Value‐at‐Risk or CAViaR) combine backtesting results with ex ante VaR estimates in an ARIMA framework in order to forecast P/L distributions more accurately. In this commentary, the authors present a systematic overview of several classes of applied statistical techniques that can make VaR backtesting more comprehensive and provide valuable insights into the analytical properties of VaR models in various market environments. In addition, they discuss the challenges associated with extending traditional backtesting approaches for VaR horizons longer than one day and propose solutions to this important problem.

This study addresses resilient mixed supply chain network design (SCND) and aims to minimize the expected total cost of the supply chain (SC) considering disruptions. The capacity of facilities is considered uncertain. In order to get closer to real-world situations, competition between SCs is considered.

A two-stage stochastic programming model is developed for designing the SC network. The location of facilities and selection of suppliers are considered first-stage decisions, and the determination of materials and products flows are second-stage decisions. Some resilience strategies are applied to mitigate the negative impacts of disruptions.

The results indicate that considering resilience and applying the related strategies are vitally important, and resilience strategies can significantly improve the SC objective and maintain market share. Also, it is confirmed that unrealistic decisions will be made without considering the competition.

This study contributes to the literature by proposing a novel mathematical model for the resilient mixed SCND problem. The other contribution is considering the chain-to-chain competition in collecting returned products and selling recycled products to other SCs in a mixed SC under disruptions. Also, a novel hybrid metaheuristic is developed to cope with the complexity of the model.