Foundations of Random-Like Bi-Level Decision Making

The investigation of bi-level (decision-making) programming problems has been strongly motivated by real world applications (Dempe, Foundations of bilevel programming. Kluwer Academic, Dordrecht, 2002). The complex nature of decision-making requires practitioners to make decisions based on a wide variety of cost considerations, benefit analyses and purely technical considerations. As we have entered the era of Big Data, which is the next frontier for innovation, competition and productivity, a new scientific revolution is about to begin. Fortunately, we will be in a position to witness the coming technological leapfrog. Many fields and sectors, ranging from economic and business activities to public administration, from national security to scientific research, are involved in Big Data problems. Because of the excessive data, inherent laws cannot be identified or the inherent data laws are changing so dynamically with “uncertainty” an outward manifestation. Therefore, practical situations are usually too complicated to be described using determinate variables and different random phenomena need to be considered for different practical problems. The development of analytical approaches, such as mathematical models with random-like uncertainty and algorithms able to solve the bi-level decision making problems are still largely unexplored. This book is an attempt to elucidate random-like bi-level decision making to all these levels. In this chapter, we present some elements of random sets theory, which includes some fundamental concepts, definitions of random variables, fuzzy variables, random-overlapped random (Ra-Ra) variables and fuzzy-overlapped random (Ra-Fu) variables. Subsequently, the elements of bi-level programming are introduced. This part is the prerequisite for reading this book. Since these results are well-known, we only introduce the results and readers can refer to correlative research such as Billingsley (Probability and measure, John Wiley & Sons, New York, 1965), Cohn (Measure theory, Birkhäuser, Boston, 1980), Halmos (Measure theory, Van Nostrad Reinhold Company, New York, 1950), and Krickeberg (Probability theory, Addison-Wesley, Reading, 1965).