The Scalar Mean Chance and Expected Value of Regular Bifuzzy Variables

As a natural extension of the fuzzy variable, a bifuzzy variable is defined as a mapping from a credibility space to the collection of fuzzy variables, which is an appropriate tool to model the two-fold fuzzy phenomena. In order to enrich its theoretical foundation, this paper explores some important measures for regular bifuzzy variables, the most commonly used type of bifuzzy variables. Firstly, we introduce the regular bifuzzy variables’ mean chance measure and some properties, including self-duality and its calculation formulas. Furthermore, we also investigate the mean chance distribution for strictly monotone functions of regular bifuzzy variables based on the proposed operational law. Finally, we present the expected value operator as well as equivalent analytical formulas of the expected value of regular bifuzzy variables and their strictly monotone functions.

OPTIMIZING RELIABILITY OF LINEAR FRACTIONAL DIFFERENCE SYSTEMS UNDER UNCERTAINTY AND RANDOMNESS

Some complex systems may suffer from failure processes arising from soft failures and hard failures. The existing researches have shown that the reliability of a dynamic system is not constant under uncertain random environments. First, two types of uncertain random optimization models are proposed where reliability index is quantified by chance measure based on whether soft and hard failures are independent or not. It is considered that internal degradation is driven by left Caputo fractional linear difference equation, while shocks are defined as discrete i.i.d. random variables. The shocks may generate additional uncertain degradation shifts when considering the competing dependent failure processes. Then, two proposed optimization reliability problems may be transformed into their equivalent deterministic forms on the basis of [Formula: see text]-path, and improved gradient descent method is applied to obtain optimal solutions. Finally, the numerical example of a micro-engine indicates that the optimization models are beneficial to the reliability of engineering systems.

16. Compensatory remedies

This chapter considers compensatory damages, the primary remedy for breach of contract which an affected party can seek in English law. It first examines how the courts assess and award damages based on two approaches: expectation interest and reliance interest. In particular, it looks at the various measures of damages such as the loss of profit measure, the ‘cost of cure’ and ‘difference in value’ measures, loss of amenity and disappointed expectations, and the loss of chance measure. The chapter also discusses the various ways in which compensatory remedies can be limited in law by focusing on the principles of causation, mitigation, and remoteness. Finally, it evaluates the role damages play in contract law and suggests that English law gives the parties a good deal of freedom to design remedies, which parties can use when designing contracts.

Bi-level chance constraint programming model for hazardous materials distribution

With the growing presence of hazardous materials in daily life, a large number of institutions and scholars have been paying close attention to this field, providing new directions for exploring hazardous materials distribution patterns. This paper employs two fuzzy random variables, transportation cost and risk, to put forward a bi-level minimum objective programming model with a chance measure constraint within a specified chance level. The lower level is to seek minimum transportation costs and the upper level is for minimum risk. The model presented in this article simultaneously designs the hybrid algorithm, which is the combination of the fuzzy random simulation with the genetic algorithm. In the end, a smallscale instance is given to account for the efficiency of the presented model and algorithm, and the best distribution solution is presented.

16. Compensatory remedies

This chapter considers compensatory damages, the primary remedy for breach of contract which an affected party can seek in English law. It first examines how the courts assess and award damages based on two approaches: expectation interest and reliance interest. In particular, it looks at the various measures of damages such as the loss of profit measure, the ‘cost of cure’ and ‘difference in value’ measures, loss of amenity and disappointed expectations, and the loss of chance measure. The chapter also discusses the various ways in which compensatory remedies can be limited in law by focusing on the principles of causation, mitigation, and remoteness. Finally, it evaluates the role damages play in contract law and suggests that English law gives the parties a good deal of freedom to design remedies, which parties can use when designing contracts.

The Economic Life Assessment of Transformer Based on Random Fuzzy Theory

According to the Life Cycle Cost theory, the paper sets up the Life Cycle Cost model of the transformer that covers initial investment cost, operation cost, failure cost and retirement cost, and puts forward a new concept of economic life, determining the economic life of transformer through comparing the annual net profit of the old transformer with the largest annual net profit of the new transformer. As the transformer serves for a long time and the running environment is complicated, parameters, which are required for in evaluation, are both random and fuzzy. Then, introduce random and fuzzy theory that can handle the uncertainties at the same time to assess the economic life of the transformer. This model not only can get the expectancy of economic life of the transformer, but also can get the economic life interval under different confidence level and the economic life interval whose chance measure is the biggest under the same economic life span. Finally, take the assessed transformer as a example to verify the rationality and validity of the model.