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Choice under risk has a large stochastic (unpredictable) component. This chapter examines five stochastic models for binary discrete choice under risk and how they combine with “structural” theories of choice under risk. Stochastic models are substantive theoretical hypotheses that are frequently testable in and of themselves, and also identifying restrictions for hypothesis tests, estimation and prediction. Econometric comparisons suggest that for the purpose of prediction (as opposed to explanation), choices of stochastic models may be far more consequential than choices of structures such as expected utility or rank-dependent utility.

In this chapter, we utilize the residual concept of productivity measures defined in the context of normal-gamma stochastic frontier production model with heterogeneity to differentiate productivity and inefficiency measures. In particular, three alternative two-way random effects panel estimators of normal-gamma stochastic frontier model are proposed using simulated maximum likelihood estimation techniques. For the three alternative panel estimators, we use a generalized least squares procedure involving the estimation of variance components in the first stage and estimated variance–covariance matrix to transform the data. Empirical estimates indicate difference in the parameter coefficients of gamma distribution, production function, and heterogeneity function variables between pooled and the two alternative panel estimators. The difference between pooled and panel model suggests the need to account for spatial, temporal, and within residual variations as in Swamy–Arora estimator, and within residual variation in Amemiya estimator with panel framework. Finally, results from this study indicate that short- and long-run variations in financial exposure (solvency, liquidity, and efficiency) play an important role in explaining the variance of inefficiency and productivity.

Fused deposition modeling (FDM) is an extensively used additive manufacturing method with the capacity to build complex functional components. Due to the machinery and environmental factors during manufacturing, the FDM parts inevitably demonstrated uncertainty in properties and performance. This study aims to identify the stochastic constitutive behaviors of FDM-fabricated polylactic acid (PLA) tensile specimens induced by the manufacturing process.

By conducting the tensile test, the effects of the printing machine selection and three major manufacturing parameters (i.e., printing speed S, nozzle temperature T and layer thickness t) on the stochastic constitutive behaviors were investigated. The influence of the loading rate was also explained. In addition, the data-driven models were established to quantify and optimize the uncertain mechanical behaviors of FDM-based tensile specimens under various printing parameters.

As indicated by the results, the uncertain behaviors of the stiffness and strength of the PLA tensile specimens were dominated by the printing speed and nozzle temperature, respectively. The manufacturing-induced stochastic constitutive behaviors could be accurately captured by the developed data-driven model with the R² over 0.98 on the testing dataset. The optimal parameters obtained from the data-driven framework were T = 231.3595 °C, S = 40.3179 mm/min and t = 0.2343 mm, which were in good agreement with the experiments.

The developed data-driven models can also be integrated into the design and characterization of parts fabricated by extrusion and other additive manufacturing technologies.

Stochastic behaviors of additively manufactured products were revealed by considering extensive manufacturing factors. The data-driven models were proposed to facilitate the description and optimization of the FDM products and control their quality.

Whereas inventory theory traditionally assumes the periodic review inventory model (R, T), with an order‐up‐to level R, has a random demand and lead time coupled with a deterministic review interval T, firms often deviate from a strict adherence to a fixed review interval when they attempt to capture transportation scale efficiencies. Employing this policy introduces additional supply chain variability. This paper aims to provide an expression for the standard deviation of demand during the protection period, important in setting safety stock, as well as an expression for the amount of order variance amplification induced by a stochastic review interval.

Analytical modeling is used to develop the expression for the standard deviation of demand during the protection period as well as the calculation for the amount of order variance amplification induced by a stochastic review interval.

In terms of the variance of demand over the protection period, a stochastic review interval has a similar effect to that of a stochastic lead time, but its impact on demand variance amplification within the supply chain differs fundamentally. Specifically, a stochastic review interval creates an order batching bullwhip effect not identified in existing literature.

This study offers an expression for the standard deviation of demand during the protection period when stochastic review intervals are employed. The expression can be used to more effectively set safety stock. The paper also offers an expression for the order variance amplification induced by a stochastic review interval.

The study offers suggestions for retailers and suppliers regarding when the use of a stochastic review interval is effective in terms of cost efficiencies.

While the existence and effect of lead time variability is well‐established in the literature, traditional approaches the periodic review inventory model ignore the stochastic nature of review interval. This paper highlights the use of stochastic review intervals as a contributing factor to the bullwhip effect.

The purpose of this paper is to analyze how exchange ratios in mergers can be assessed when the companies economic capital valuation is carried out in a stochastic framework with financial assets and minimum guarantees.

The paper is a theoretical one. Its main objective is to present a quantitative model for exchange ratios accounting, introducing a stochastic pricing model in the presence of stochastic cash‐flows and representing contractual embedded real option such as minimum guarantees.

The paper presents a financial model to evaluate the differences in exchange ratios induced by stochastic capital reserves in the merging companies.

Stochastic cash‐flows in the economic capital of the merging companies set up a stochastic capital reserve which represents an additional value and could induce important differences in exchange ratios.

The model is fully applicable, also in the presence of embedded real options such as minimum guarantees, but requires the volatility of the underlying.

The paper should be useful under both a managerial and a theoretical use in order to evaluate stochastic exchange ratios.

With respect to multiple attribute group decision-making (MAGDM) in which the assessment values of alternatives are denoted by normal discrete fuzzy variables (NDFVs) and the weight information of attributes is incompletely known, this paper aims to develop a novel fuzzy stochastic MAGDM method based on credibility theory and fuzzy stochastic dominance, and then applies the proposed method for selecting the most desirable investment alternative under uncertain environment.

First, by aggregating the membership degrees of an alternative to a scale provided by all decision-makers into a triangular fuzzy number, the credibility degree and expect the value of a triangular fuzzy number are calculated to construct the group fuzzy stochastic decision matrix. Second, based on determining the credibility distribution functions of NDFVs, the fuzzy stochastic dominance relations between alternatives on each attribute are obtained and the fuzzy stochastic dominance degree matrices are constructed by calculating the dominance degrees that one alternative dominates another on each attribute. Subsequently, calculating the overall fuzzy stochastic dominance degrees of an alternative on each attribute, a single objective non-linear optimization model is established to determine the weights of attributes by maximizing the relative closeness coefficients of all alternatives to positive ideal solution. If the information about attribute weights is completely unknown, the idea of maximizing deviation is used to determine the weights of attributes. Finally, the ranking order of alternatives is determined according to the descending order of corresponding relative closeness coefficients and the best alternative is determined.

This paper proposes a novel fuzzy stochastic MAGDM method based on credibility theory and fuzzy stochastic dominance, and a case study of investment alternative selection problem is provided to illustrate the applicability and sensitivity of the proposed method and its effectiveness is demonstrated by comparison analysis with the proposed method with the existing fuzzy stochastic MAGDM method. The result shows that the proposed method is useful to solve the MAGDM problems in which the assessment values of alternatives are denoted by NDFVs and the weight information of attributes is incompletely known.

The contributions of this paper are that to describe the dominance relations between fuzzy variables reasonably and quantitatively, the fuzzy stochastic dominance relations between any two fuzzy variables are redefined and the concept of fuzzy stochastic dominance degree is proposed to measure the dominance degree that one fuzzy variable dominate another; Based on credibility theory and fuzzy stochastic dominance, a novel fuzzy stochastic MAGDM method is proposed to solve MAGDM problems in which the assessment values of alternatives are denoted by NDFVs and the weight information of attributes is incompletely known. The proposed method has a clear logic, which not only can enrich and develop the theories and methods of MAGDM but also provides decision-makers a novel method for solving fuzzy stochastic MAGDM problems.

The purpose of this paper is to research stochastic dynamic investment games with stochastic interest rate model in continuous time between two investors. The market interest rate has the dynamics of Duffie‐Kan interest rate.

Recently, there has been an increasing interest in financial market models whose key parameters, such as the bank interest rate, stocks appreciation rates, and volatility rates, are modulated by stochastic interest rate. This paper uses the Duffie‐Kan stochastic interest rate model to develop stochastic differential portfolio games. By the HJB optimality equation, a general result in optimal control for a stochastic differential game with a general utility payoff function is obtained.

Derive a general result in optimal control for a stochastic differential game with a general utility payoff function. The explicit optimal strategies and value of the games are obtained for the constant relative risk aversion utility games of fixed duration.

Accessibility and availability of stochastic interest rate data are the main limitations, which apply.

The results obtained in this paper could be used as a guide to actual portfolio games.

This paper presents a new approach for the optimal portfolio model under compound jump processes. The paper is aimed at actual portfolio games.

The purpose of this paper is to implement the hardware structure for radial basis function (RBF) neural network based on stochastic logic computation.

The hardware implementation of artificial neural networks (ANNs) has a complicated structure and is normally space consuming due to huge size of digital multiplication, addition/subtraction, non‐linear activation function, etc. Also the unavailability of ANN hardware at an attractive price limits its use for real time applications. In stochastic logic theory, the real numbers are converted to random streams of bits instead of a binary number. The performance of the proposed structure is analyzed using very high speed integrated circuit hardware description language.

Stochastic theory‐based arithmetic and logic approach provides a way to carry out complex computation with very simple hardware and very flexible design of the system. The Gaussian RBF for hidden layer neuron is employed using stochastic counter that reduces the hardware resources significantly. The number of hidden layer neurons in RBF neural network structure is adaptively varied to make it an intelligent system.

The paper outlines the stochastic neural computation on digital hardware for implementing radial basis neural network. The structure has considered the optimized usage of hardware resources.

This paper aims to present two-stage network models in the presence of stochastic ratio data.

Black-box, free-link and fix-link techniques are used to apply the internal relations of the two-stage network. A deterministic linear programming model is derived from a stochastic two-stage network data envelopment analysis (DEA) model by assuming that some basic stochastic elements are related to the inputs, outputs and intermediate products. The linkages between the overall process and the two subprocesses are proposed. The authors obtain the relation between the efficiency scores obtained from the stochastic two stage network DEA-ratio considering three different strategies involving black box, free-link and fix-link. The authors applied their proposed approach to 11 airlines in Iran.

In most of the scenarios, when alpha in particular takes any value between 0.1 and 0.4, three models from Charnes, Cooper, and Rhodes (1978), free-link and fix-link generate similar efficiency scores for the decision-making units (DMUs), While a relatively higher degree of variations in efficiency scores among the DMUs is generated when the alpha takes the value of 0.5. Comparing the results when the alpha takes the value of 0.1–0.4, the DMUs have the same ranking in terms of their efficiency scores.

The authors innovatively propose a deterministic linear programming model, and to the best of the authors’ knowledge, for the first time, the internal relationships of a two-stage network are analyzed by different techniques. The comparison of the results would be able to provide insights from both the policy perspective as well as the methodological perspective.