Fundamental Concepts

Fundamental Concepts

Copyright: © 2019 |Pages: 51
DOI: 10.4018/978-1-5225-8301-1.ch002

Abstract

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.
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2.1 Elementary Idea On Probability Theory

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.

  • Definition 2.1.1: (978-1-5225-8301-1.ch002.m01978-1-5225-8301-1.ch002.m02Algebra): Let 978-1-5225-8301-1.ch002.m03 be a random experiment and 978-1-5225-8301-1.ch002.m04 be its sample space. A non-empty collection 978-1-5225-8301-1.ch002.m05 of subsets of 978-1-5225-8301-1.ch002.m06 is said to form an 978-1-5225-8301-1.ch002.m07978-1-5225-8301-1.ch002.m08 algebra, if the following conditions are satisfied

    • o

      if 978-1-5225-8301-1.ch002.m09, then 978-1-5225-8301-1.ch002.m10.

    • o

      if 978-1-5225-8301-1.ch002.m11, then 978-1-5225-8301-1.ch002.m12.

  • Definition 2.1.2: (Probability Space): Let 978-1-5225-8301-1.ch002.m13 be a 978-1-5225-8301-1.ch002.m14978-1-5225-8301-1.ch002.m15 algebra. A mapping 978-1-5225-8301-1.ch002.m16978-1-5225-8301-1.ch002.m17 that maps each element of 978-1-5225-8301-1.ch002.m18to a real number978-1-5225-8301-1.ch002.m19, is called probability measure. If 978-1-5225-8301-1.ch002.m20 is a sample space, then the triple 978-1-5225-8301-1.ch002.m21is called a probability space.

  • Definition 2.1.3: (Random Variables): Let 978-1-5225-8301-1.ch002.m22be a random experiment and 978-1-5225-8301-1.ch002.m23 be its sample space. If for each event point 978-1-5225-8301-1.ch002.m24 of the sample space 978-1-5225-8301-1.ch002.m25, there is a real number 978-1-5225-8301-1.ch002.m26 by a given rule, i.e., if 978-1-5225-8301-1.ch002.m27 is a mapping from the sample space 978-1-5225-8301-1.ch002.m28 to the set of real numbers 978-1-5225-8301-1.ch002.m29, i.e., 978-1-5225-8301-1.ch002.m30, such that 978-1-5225-8301-1.ch002.m31 for all 978-1-5225-8301-1.ch002.m32, then 978-1-5225-8301-1.ch002.m33 is called a random variable or a stochastic variable.

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