Fuzzy Bi-level Decision-Making Techniques: A Survey <s> Basic Fuzzy Bi-level Decision-Making <s> The focus of this book is on bilevel programming which combines elements of hierarchical optimization and game theory. The basic model addresses the problem where two decision-makers, each with their individual objectives, act and react in a noncooperative manner. The actions of one affect the choices and payoffs available to the other but neither player can completely dominate the other in the traditional sense. Over the last 20 years there has been a steady growth in research related to theory and solution methodologies for bilevel programming. This interest stems from the inherent complexity and consequent challenge of the underlying mathematics, as well as the applicability of the bilevel model to many real-world situations. The primary aim of this book is to provide a historical perspective on algorithmic development and to highlight those implementations that have proved to be the most efficient in their class. A corollary aim is to provide a sampling of applications in order to demonstrate the versatility of the basic model and the limitations of current technology. What is unique about this book is its comprehensive and integrated treatment of theory, algorithms and implementation issues. It is the first text that offers researchers and practitioners an elementary understanding of how to solve bilevel programs and a perspective on what success has been achieved in the field. Audience: Includes management scientists, operations researchers, industrial engineers, mathematicians and economists. <s> BIB001 </s> Fuzzy Bi-level Decision-Making Techniques: A Survey <s> Basic Fuzzy Bi-level Decision-Making <s> Bilevel decision techniques are mainly developed for solving decentralized management problems with decision makers in a hierarchical organization. Organizational bilevel decision-making, such as planning of land-use, transportation and water resource, all may involve uncertain factors. The parameters shown in a bileved programming model, either in the objective functions or constraints, are thus often imprecise, which is called fuzzy parameter bilevel programming (FPBLP) problem. Following our previous work (1, 2), this study first proposes a model of FPBLP. It then gives the definition of optimal solution for an FPBLP problem. Based on the definition and related theorems, this study develops a fuzzy number based Kuhn-Tucher approach to solve the proposed FPBLP problem. Finally, an example further illustrates the power of the fuzzy number based Kuhn-Tucher approach. <s> BIB002 </s> Fuzzy Bi-level Decision-Making Techniques: A Survey <s> Basic Fuzzy Bi-level Decision-Making <s> Purpose – This study aims to develop a decision making model and approach for logistics planning problem which naturally involves two or more decision units at a hierarchical structure. Such a decision problem in practice often involves uncertain and imprecise factors with the parameters of a bilevel decision model, either in the objective functions or constraints.Design/methodology/approach – This paper proposes a fuzzy bilevel decision making model for a general logistics planning problem and develops a fuzzy number based Kth‐best approach to find an optimal solution for the proposed fuzzy bilevel decision problem.Findings – The proposed approach illustrates an optimal solution in logistics management, which meets maximally/minimally the objectives of both supplier and distributor (or other parts of the logistics chain). The proposed fuzzy bilevel decision approach can have a wide range of logistics management applications.Research limitations/implications – The decision model, approach and system will ... <s> BIB003 </s> Fuzzy Bi-level Decision-Making Techniques: A Survey <s> Basic Fuzzy Bi-level Decision-Making <s> In this paper, a fuzzy goal programming (FGP) model is developed to solve quadratic bilevel programming problems with fuzzy resource constraints for proper distribution of decision powers to the leader and follower. In the model, formulation of the problem, concept of tolerance membership functions for measuring the degree of satisfaction of the objective of the leader and follower are defined first, under the fuzzily described system constraints. Subsequently, a quadratic programming model is constructed on the basis of degree of satisfaction of both the leader and follower. The developed model is converted into an equivalent non-linear FGP model to achieve the highest degree of satisfaction (unity) to the extent possible. In the decision process, the Taylor's series approximation technique is applied to linearise the non-linear goals and to achieve the fuzzy goal objective values of the decision-makers at both the levels, by arriving at most satisfactory solution regarding the optimality of two different sets of decision variables controlled separately by each of them. An illustrative example is solved to demonstrate the efficiency of the proposed approach and the solution is compared with the solution obtained by using an existing methodology developed by Osman et al. in 2004. <s> BIB004 </s> Fuzzy Bi-level Decision-Making Techniques: A Survey <s> Basic Fuzzy Bi-level Decision-Making <s> This article considers bilevel linear programming problems where the coefficients of the objective functions and the constraints in the problem are given as fuzzy parameters. Stackelberg solutions under fuzziness are defined by incorporating the notions of possibility theory into the original concept of Stackelberg solutions. It is shown that Stackelberg problems under fuzziness are transformed into deterministic bilevel linear or nonlinear programming problems, and that the resulting problems are exactly solved by using conventional bilevel linear or nonlinear programming techniques. <s> BIB005 </s> Fuzzy Bi-level Decision-Making Techniques: A Survey <s> Basic Fuzzy Bi-level Decision-Making <s> In this paper, we develop a fuzzy tri-level decision making (FTLDM) model to deal with decentralized decision making problems with three levels of decision makers. Based on the λ-cut of fuzzy set, we transform an FTLDM problem into a multiobjective tri-level decision making problem. Based on the linear tri-level Kth-best algorithm, the global optimal solution can be obtained. A case study for third-party logistics decision making in supply chain is utilized to illustrate the effectiveness of the proposed algorithm. <s> BIB006 </s> Fuzzy Bi-level Decision-Making Techniques: A Survey <s> Basic Fuzzy Bi-level Decision-Making <s> In the present paper, a solution approach to fuzzy bilevel optimization problem is given for the linear case. We use -cuts of the fuzzy polytopes to attack the fuzzy optimization problem on upper level. Then, crisp optimization problems arise. Thus, the solution of the fuzzy upper level problem turns out to be a union of the solutions of these crisp problems on all -cuts. The solution approach for the lower level fuzzy optimization problem was investigated by the author in previous publications and is implemented in the present paper. <s> BIB007
This section reviews the development of techniques for solving a category of fuzzy bi-level decision-making problems in which only one decision entity is involved at each decision level that attempts to optimize a unique objective; this category of problems can be termed basic fuzzy bi-level decision problems. A general fuzzy linear bi-level decision problem (in the linear form) can be described as Definition 1. , a general fuzzy linear bilevel decision problem is defined as follows: where, for each x given by the leader, y solves(1c-1d) where BIB001 . The original appearance of fuzzy bi-level decisionmaking was proposed by Sakawa et al.  on a linear bi-level decision model with fuzzy parameters. The authors developed an interactive fuzzy programming approach for solving the problem. They then extended the interactive fuzzy programming approach to solve fuzzy linear bi-level fractional decision problems  . Sakawa, et al.   also used the interactive fuzzy programming approach and genetic algorithms to solve fuzzy bi-level 0-1 decision problems and fuzzy bi-level nonconvex decision problems. In recent years, Pramanik  proposed a fuzzy goal programming approach to solve fuzzy linear bi-level decision problems. Biswas and Bose BIB004 developed a fuzzy goal programming approach for solving quadratic bi-level decision problems with fuzzy parameters that are only involved in constraint conditions. In general, these fuzzy approaches can be used to find solutions to a range of fuzzy bi-level decision problems. However, these methods are limited to solving fuzzy bi-level decision problems in special situations where decision entities from different levels share the same constraint conditions and prefer to cooperate with one another. Many researchers extended traditional bi-level programming methods to solve fuzzy bi-level decision problems. Zhang and Lu BIB002 developed an approximation Kuhn-Tucker approach to solve fuzzy linear bi-level decision problems. They also presented an approximation Kth-Best algorithm and an approximation branch and bound algorithm to solve the fuzzy linear bi-level decision problem BIB003  . Gao et al.  proposed a programmable λ-cut approximation algorithm to solve a λ-cut set based fuzzy goal bi-level decision problem. Budnitzki BIB007 used the selection function approach and a modified version of the KthBest algorithm to solve a fuzzy linear bi-level decision problem. Katagiri et al. BIB005 defined Stackelberg solutions for fuzzy linear bi-level decision problems and then transformed fuzzy problems into deterministic linear or nonlinear bi-level decision problems that can be exactly solved by conventional bi-level solution methods. Furthermore, Zhang et al. BIB006 presented a fuzzy linear tri-level decision problem, and extended the linear tri-level Kth-Best algorithm proposed by Zhang et al.  to solve the problem based on λ-cut of a fuzzy set. Clearly, these solution methods are mainly for solving linear form of fuzzy bi-level and tri-level decision-making.