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The purpose of this work is to provide theoretical foundations of, as well as some computational aspects on, a theory for analysing decisions under risk, when the available information is vague and imprecise. Many approaches to model unprecise information, e.g., by using interval methods, have prevailed. However, such representation models are unnecessarily restrictive since they do not admit discrimination between beliefs in different values, i.e., the epistemologically possible values have equal weights. In many situations, for instance, when the underlying information results from learning techniques based on variance analyses of statistical data, the expressibility must be extended for a more perceptive treatment of the decision situation. Our contribution herein is an approach for enabling a refinement of the representation model, allowing for an elaborated discrimination of possible values by using belief distributions with weak restrictions. We show how to derive admissible classes of local distributions from sets of global distributions and introduce measures expressing into which extent explicit local distributions can be used for modelling decision situations. As will turn out, this results in a theory that has very attractive features from a computational viewpoint.

The paper offers a new perspective on optimal portfolio choice by investigating how and to what extent knowledge of an investor's desirable initial investment choice can be used to determine his future optimal portfolio allocations. Optimality of investment decisions is built on the so-called forward investment performance criteria and, in particular, on the time-monotone ones. It is shown that for this class of forward criteria the desired initial allocations completely characterize the future optimal investment strategies. The analysis uses the connection between a nonlinear equation, satisfied by the local risk tolerance, and the backward heat equation. Complete solutions are provided as well as various examples.

One of the most complex systems is the human brain whose formalized functioning is characterized by decision theory. We present a "Quantum Decision Theory" of decision-making, based on the mathematical theory of separable Hilbert spaces. This mathematical structure captures the effect of superposition of composite prospects, including many incorporated intentions, which allows us to explain a variety of interesting fallacies and anomalies that have been reported to particularize the decision-making of real human beings. The theory describes entangled decision-making, non-commutativity of subsequent decisions, and intention interference of composite prospects. We demonstrate how the violation of the Savage's sure-thing principle (disjunction effect) can be explained as a result of the interference of intentions, when making decisions under uncertainty. The conjunction fallacy is also explained by the presence of the interference terms. We demonstrate that all known anomalies and paradoxes, documented in the context of classical decision theory, are reducible to just a few mathematical archetypes, all of which allow the finding of straightforward explanations in the frame of the developed quantum approach.

In this paper, we perform the analysis of temporalized structure of a body of evidence and possibilistic Extremal Fuzzy Dynamic System (EFDS) for the construction of more precise decisions based on the expert knowledge stream. The process of decision precision consists of two stages. In the first stage the relation of information precision is defined on a monotone sequence of bodies of evidence. The principle of negative imprecision is developed, as the maximum principle of knowledge ignorance measure of a body of evidence. Corresponding mathematical programming problem is constructed. On the output of the first stage we receive the expert knowledge precision stream of the criteria with respect to any decision. In the second stage the constructed stream is an input trajectory for the finite possibilistic model of EFDS. A genetic algorithm approach is developed for identifying of the EFDS finite model. The modelling process gives us the more precise decisions as a prediction of a temporalization procedure. The constructed technology is applied in the non-probabilistic utility theory for the information technology management problem.

Surface finish is the most desired properties of any sophisticated machinery parts for its proper functioning and long endurance. The surface finish must be in the order of micrometer to nanometer for most of the machinery parts. The nontraditional surface finish processes prepare these parts; in these processes, the uses of electromagnet play a vital role in the surface finishing mechanism. Magnetic abrasive flow finishing (MAFF) is such a hybrid process, which gives a combined effect of abrasive flow finishing (AFF) and magnetic abrasive finishing (MAF). In this method, a pair of electromagnets are attached to the AFF setup. By using electromagnet in the AFF process, it enhanced material removal and surface finishing. The main process parameters selected in the MAFF process were magnetic flux density, number of cycles, percentage abrasive content, piston speed, and corresponding responses selected were material removal, percentage improvement in surface finish. In this research paper, the responses were optimized by a combination of utility theory and meta-heuristic firefly’s algorithm. The utility theory based-firefly algorithm’s predicted global optimum parameters set, which was more suitable for reducing the finishing time and required surface finish. The confirmatory test validated this optimized parameter set and it was revealed that the meta-heuristic firefly algorithm embedded with utility theory had given optimized results in the MAFF process.

It is well known that statistics deals with quantitative analysis. Thus, there is a lack of approach to do quantitative analysis in the presence of qualitative attributes. Soft set theory has the freedom to deal with attributes, [0, 1], etc. along with quantity. Thus, we introduce some fundamental ideas of soft statistics.

Here, soft mean, soft standard deviation, soft coefficient of variation, soft correlation coefficient are introduced and some theorems are proved with respect to utility. Utility theory provides an analysis of choice behavior. As an application of our notions and results, we find soft correlation coefficients between vulnerability and government responses of various regions across the world. The data from “The Global Slavery Index 2016” are considered for application purposes.

In this chapter, we first discuss utility theory and utility function in detail, then we show how asset allocation can be done in terms of the quadratic utility function. Based upon these concepts, we show Markowitz’s portfolio selection model can be executed by constrained maximization approach. Real world examples in terms of three securities are also demonstrated. In the Markowitz selection model, we consider that short sale is both allowed and not allowed.

The main purposes of this introduction chapter are (i) to give an overview of the following 109 papers, which discuss investment analysis, portfolio management, and financial derivatives; (ii) to classify these 109 chapters into nine topics; and (iii) to classify the keywords in terms of chapter numbers.

In this chapter, we first discuss utility theory and utility function in detail, then we show how asset allocation can be done in terms of quadratic utility function. Based upon these concepts, we show that Markowitz’s portfolio selection model can be executed by the constrained maximization approach. Real-world examples in terms of three securities are also demonstrated. In the Markowitz selection model, we consider that short sale is both allowed and not allowed.

The paper offers a new perspective on optimal portfolio choice by investigating how and to what extent knowledge of an investor's desirable initial investment choice can be used to determine his future optimal portfolio allocations. Optimality of investment decisions is built on the so-called forward investment performance criteria and, in particular, on the time-monotone ones. It is shown that for this class of forward criteria the desired initial allocations completely characterize the future optimal investment strategies. The analysis uses the connection between a nonlinear equation, satisfied by the local risk tolerance, and the backward heat equation. Complete solutions are provided as well as various examples.

This paper surveys the most recent advances in the context of decision making under uncertainty, with an emphasis on the modeling of risk-averse preferences using the apparatus of axiomatically defined risk functionals, such as coherent measures of risk and deviation measures, and their connection to utility theory, stochastic dominance, and other more established methods.