Bi-Level Decision Making in Ra-Fu Phenomenon

In the real-life world, there are two kinds of common uncertainties, i.e., randomness and fuzziness. Accordingly, two efficient theories, probability theory and possibility theory are developed to handle them. For the possibility theory, readers may refer to Dubois and Prade (Possibility theory: an approach to computerized processing of uncertainty. Plenum Press, New York, 1988), Dubois and Prade (Possibility theory: qualitative and quantitative aspects. In: Quantified representation of uncertainty and imprecision. Springer, Berlin/New York, pp 169–226, 1998), and Zadeh (Fuzzy Sets Syst 1(1):3–28, 1978). In some decision problems such as supply chain network problems (Xu et al., Inf Sci 178(8):2022–2043, 2008), inventory problems (Xu and Liu, Inf Sci 178(14):2899–2914, 2008), portfolio selection problems (Li and Xu, Omega 37(2):439–449, 2009), the mixture of randomness and fuzziness is required to be considered simultaneously. From a viewpoint of ambiguity and randomness different from fuzzy random variables (Krätschmer V, Fuzzy Sets Syst 123(1):1–9, 2001; Kruse and Meyer, Statistics with vague data. Springer, Dordrecht, 1987; Kwakernaak, Inf Sci 15(1):1–29, 1978; Puri and Ralescu, J Math Anal Appl 114(2):409–422, 1986), by considering the experts’ ambiguous understanding of means and variances of random variables, a concept of random fuzzy variables is proposed. In this chapter, we consider the construction site security planning (CSSP) problems with Ra-Fu phenomenon. Then three classes of bi-level mathematical models with Ra-Fu parameters are developed. Some theoretical results and algorithms are proposed for the models. At the end of the chapter, a practical case study demonstrates the feasibility and efficiency of the proposed methods.