Fuzzy Bi-level Decision-Making Techniques: A Survey <s> Fuzzy Random Bi-level Decision-Making <s> Fuzziness is discussed in the context of multivalued logic, and a corresponding view of fuzzy sets is given. Fuzzy random variables are introduced as random variables whose values are not real but fuzzy numbers, and subsequently redefined as a particular kind of fuzzy set. Expectations of fuzzy random variables, characteristic functions of fuzzy events, probabilities connected to fuzzy random variables, and conditional expectations and probabilities relating to fuzzy random variables are defined as images of the fuzzy set representing the fuzzy random variable under appropriate mappings. Several theorems, some of which relate to independent fuzzy random variables, are proved. <s> BIB001 </s> Fuzzy Bi-level Decision-Making Techniques: A Survey <s> Fuzzy Random Bi-level Decision-Making <s> There are two important sources of uncertainty: randomness and fuzziness. Randomness models the stochastic variability of all possible outcomes of a situation, and fuzziness relates to the unsharp boundaries of the parameters of the model. In this sense, randomness is largely an instrument of a normative analysis that focuses on the future, while fuzziness is more an instrument of a descriptive analysis reflecting the past and its implications. Clearly, randomness and fuzziness are complementary, and so a natural question is how fuzzy variables could interact with the type of random variables found in actuarial science. This article focuses on one important dimension of this issue, fuzzy random variables (FRVs). The goal is to introduce IME readers to FRVs and to illustrate how naturally compatible and complementary randomness and fuzziness are. <s> BIB002 </s> Fuzzy Bi-level Decision-Making Techniques: A Survey <s> Fuzzy Random Bi-level Decision-Making <s> In this paper, assuming noncooperative behavior of the decision makers, solution methods for decision-making problems in hierarchical organizations under fuzzy random environments are presented. Taking into account the vagueness of judgments of decision makers, fuzzy goals are introduced into the formulated fuzzy random noncooperative bilevel linear programming problems. Considering the possibility and necessity measure that each objective function fulfills, the corresponding fuzzy goal, the fuzzy random bilevel linear programming problems to minimize each objective function with fuzzy random variables, are transformed into stochastic bilevel programming problems to maximize the degree of possibility and necessity that each fuzzy goal is fulfilled. Through expectation optimization in stochastic programming, which is suitable for risk-neutral decision makers, the transformed stochastic bilevel programming problems can be reduced to deterministic bilevel programming problems. For the transformed problems, extended concepts of Stackelberg solutions are introduced and computational methods are also presented. It is shown that the extended Stackelberg solutions can be obtained through the combined use of the variable transformation method and the Kth best algorithm for bilevel linear programming problems. A numerical example is provided to illustrate the proposed methods. <s> BIB003 </s> Fuzzy Bi-level Decision-Making Techniques: A Survey <s> Fuzzy Random Bi-level Decision-Making <s> This article considers bilevel linear programming problems where random fuzzy variables are contained in objective functions and constraints. In order to construct a new optimization criterion under fuzziness and randomness, the concept of value at risk and possibility theory are incorporated. The purpose of the proposed decision making model is to optimize possibility-based values at risk. It is shown that the original bilevel programming problems involving random fuzzy variables are transformed into deterministic problems. The characteristic of the proposed model is that the corresponding Stackelberg problem is exactly solved by using nonlinear bilevel programming techniques under some convexity properties. A simple numerical example is provided to show the applicability of the proposed methodology to real-world hierarchical problems. <s> BIB004 </s> Fuzzy Bi-level Decision-Making Techniques: A Survey <s> Fuzzy Random Bi-level Decision-Making <s> This paper considers a class of bilevel linear programming problems in which the coefficients of both objective functions are fuzzy random variables. The main idea of this paper is to introduce the Pareto optimal solution in a multi-objective bilevel programming problem as a solution for a fuzzy random bilevel programming problem. To this end, a stochastic interval bilevel linear programming problem is first introduced in terms of α-cuts of fuzzy random variables. On the basis of an order relation of interval numbers and the expectation optimization model, the stochastic interval bilevel linear programming problem can be transformed into a multi-objective bilevel programming problem which is solved by means of weighted linear combination technique. In order to compare different optimal solutions depending on different cuts, two criterions are given to provide the preferable optimal solutions for the upper and lower level decision makers respectively. Finally, a production planning problem is given to demonstrate the feasibility of the proposed approach. <s> BIB005
Fuzziness and randomness are two main sources of uncertainty BIB002 . Bi-level decision-making may appear in not only a fuzzy situation but also a random environment, and this is known as fuzzy random bilevel decision-making. Related parameters in relation to fuzzy random bi-level decision-making are called fuzzy random variables, which are introduced as random variables whose values are not real, but fuzzy numbers, and subsequently redefined as a particular kind of fuzzy sets BIB001 . With respect to Definition 1, if the coefficients of decision variables are characterized by fuzzy random variables, the fuzzy bi-level program (1) is known as a fuzzy random bi-level decision problem. In recent years, attention has been increasingly paid to fuzzy random bi-level decision-making. Sakawa et al. [42] considered a linear bi-level decision problem under a fuzzy random environment and transformed the problem into a deterministic bi-level decision problem through level sets and probability maximization for finding a Stackelberg solution. Sakawa et al. BIB003 also used another different method based on expectation optimization using possibility and necessity to reduce a fuzzy random linear bi-level decision problem to a deterministic bi-level decision problem for solving. Ren and Wang  transformed a fuzzy random linear bilevel decision problem into a deterministic bi-level multi-objective decision problem using level sets and expectation optimization, and found a optimistic Stackelberg solution to the considered problem. Ren et al.  and Ren and Wang BIB005 developed interval programming approaches for solving fuzzy random linear bi-level decision problems. Katagiri et al. BIB004 introduced the concept of value at risk and possibility theory, and transformed a fuzzy random bi-level decision problem into a set of deterministic problems that can be solved by nonlinear bi-level decisionmaking techniques.