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Biswas, Animesh and Arnab Kumar De. "Fundamental Concepts." Multi-Objective Stochastic Programming in Fuzzy Environments. IGI Global, 2019. 27-77. Web. 13 Dec. 2019. doi:10.4018/978-1-5225-8301-1.ch002

APA

Biswas, A., & De, A. K. (2019). Fundamental Concepts. In Multi-Objective Stochastic Programming in Fuzzy Environments (pp. 27-77). Hershey, PA: IGI Global. doi:10.4018/978-1-5225-8301-1.ch002

Chicago

Biswas, Animesh and Arnab Kumar De. "Fundamental Concepts." In Multi-Objective Stochastic Programming in Fuzzy Environments, 27-77 (2019), accessed December 13, 2019. doi:10.4018/978-1-5225-8301-1.ch002

In this chapter, the authors discuss some basic concepts of probability theory and possibility theory that are useful when reading the subsequent chapters of this book. The multi-objective fuzzy stochastic programming models developed in this book are based on the concepts of advanced topics in fuzzy set theory and fuzzy random variables (FRVs). Therefore, for better understanding of these advanced areas, the authors at first presented some basic ideas of probability theory and probability density functions of different continuous probability distributions. Afterwards, the necessity of the introduction of the concept of fuzzy set theory, some important terms related to fuzzy set theory are discussed. Different defuzzification methodologies of fuzzy numbers (FNs) that are useful in solving the mathematical models in imprecisely defined decision-making environments are explored. The concept of using FRVs in decision-making contexts is defined. Finally, the development of different forms of fuzzy goal programming (FGP) techniques for solving multi-objective decision-making (MODM) problems is underlined.

FRVs play an important role in developing multi-objective fuzzy stochastic programming models. To get a brief concept on FRVs, the concept of random variables is highly needed by the readers. From that view point a short discussion on basic probability theory with various types of probability distributions are briefly highlighted at first in this section, so that the readers can easily capture the idea of FRVs and can distinguish the difference between random variables and FRVs.

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Definition 2.1.1: (Algebra): Let be a random experiment and be its sample space. A non-empty collection of subsets of is said to form an algebra, if the following conditions are satisfied

o

if , then .

o

if , then .

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Definition 2.1.2: (Probability Space): Let be a algebra. A mapping that maps each element of to a real number, is called probability measure. If is a sample space, then the triple is called a probability space.

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Definition 2.1.3: (Random Variables): Let be a random experiment and be its sample space. If for each event point of the sample space , there is a real number by a given rule, i.e., if is a mapping from the sample space to the set of real numbers , i.e., , such that for all , then is called a random variable or a stochastic variable.