Fuzzy Bi-level Decision-Making Techniques: A Survey <s> Introduction <s> Abstract : A model is presented which enables an oligopolist to maximize profits subject to resource constraints and to constraints on maintenance of market share in the face of collusive action by competitors. An example is formulated and is solved using recently developed optimization methods. (Author) <s> BIB001 </s> Fuzzy Bi-level Decision-Making Techniques: A Survey <s> Introduction <s> This paper contains a bibliography of all references central to bilevel and multilevel programming that the authors know of. It should be regarded as a dynamic and permanent contribution since all the new and appropriate references that are brought to our attention will be periodically added to this bibliography. Readers are invited to suggest such additions, as well as corrections or modifications, and to obtain a copy of the LaTeX and BibTeX files that constitute this manuscript, using the guidelines contained in this paper. <s> BIB002 </s> Fuzzy Bi-level Decision-Making Techniques: A Survey <s> Introduction <s> Within the framework of any bilevel decision problem, a leader's decision is influenced by the reaction of his or her follower. When multiple followers who may have had a share in decision variables, objectives and constraints are involved in a bilevel decision problem, the leader's decision will be affected, not only by the reactions of these followers, but also by the relationships among these followers. This paper firstly identifies nine different kinds of relationships (S"1 to S"9) amongst followers by establishing a general framework for bilevel multi-follower decision problems. For each of the nine a corresponding bilevel multi-follower decision model is then developed. Also, this paper particularly proposes related theories focusing on an uncooperative decision problem (i.e., S"1 model), as this model is the most basic one for bilevel multi-follower decision problems over the nine kinds of relationships. Moreover, this paper extends the Kuhn-Tucker approach for driving an optimal solution from the uncooperative decision model. Finally, a real case study of a road network problem illustrates the application of the uncooperative bilevel decision model and the proposed extended Kuhn-Tucker approach. <s> BIB003 </s> Fuzzy Bi-level Decision-Making Techniques: A Survey <s> Introduction <s> In a trilevel hierarchical decision problem, the objectives and variables of each decision entity at one level are controlled, in part, by the decision entities at other levels. The choice of values for the decision variables at each level may influence the decisions made at other levels, and may thereby improve/reduce the objective for each level. When multiple decision entities are involved at the middle and bottom levels of a trilevel decision problem, the top-level entity's decision will be affected, not only by these followers' individual reactions, but also by the relationships between them. We call this problem a multifollower trilevel (MFTL) decision. This paper firstly defines and analyzes various kinds of relationships between decision entities in an MFTL decision problem. We then propose an MFTL decision making framework, in which 64 standard MFTL decision situations and their possible combinations are identified. To model these MFTL decision situations, we developed an innovative decision entity-relationship diagram (DERD) approach. We also established a general model for MFTL decision making and a set of standard MFTL decision models using trilevel programming. A trilevel decision support system (TLDSS) software has also been developed to transfer a DERD into a programming model. Finally, a case study illustrates typical MFTL decision making models and their development, using both DERD and programming approaches. <s> BIB004 </s> Fuzzy Bi-level Decision-Making Techniques: A Survey <s> Introduction <s> Decision-making problems in decentralised organisations are often modelled as Stackelberg games, and they are formulated as two-level mathematical programming problems with two decision-makers. If they do not have any motivation to cooperate mutually and behave rationally, outcomes of the problems can be explained by Stackelberg equilibrium which is not always Pareto optimal. From computational aspects, it is known that solving two-level programming problems is NP-hard even if the objective functions and the constraint functions are linear. In contrast, if the decision-makers can select strategies cooperatively, the most important aspect is to derive a Pareto optimal solution favourable to the decision-makers. As a method of this line of approach interactive fuzzy programming has been developed, taking into account fuzziness of human judgements. In this paper, after reviewing the development of solution methods for two- and multi-level programming problems, we focus on cooperative decision-making in decentralised organisations and give interactive fuzzy programming for two-level linear programming problems, which provides satisfactory solutions in accordance with the preference of the decision-makers. Moreover, we present extensions of interactive fuzzy programming for two-level linear programming problems under multi-objective environments and under uncertainty. <s> BIB005
Bi-level decision-making techniques, mainly implemented by bi-level programming, motivated by Stackelberg game theory  , have been developed to address compromises between two interactive decision entities that are distributed throughout a hierarchical organization. In a bi-level decision-making process, decision entities at the upper level and the lower level are respectively termed the leader and the follower, and make their individual decisions in sequence with the aim of optimizing their respective objectives. The original appearance of bi-level decision-making was proposed by Bracken and McGill BIB001 . A wide range of research in this area has been undertaken with the designations bi-level programming, bi-level optimization, and bi-level decision-making. Bi-level decision-making mainly is a bi-level optimization issue and mainly applies bi-level programming methods. It often appears in many decentralized management problems, and has motivated a number of researches to work on bi-level decision models BIB004 BIB003 , solution approaches BIB005  and applications   . A challenge in handling bi-level decision problems is that various uncertainties (including fuzziness and randomness) have naturally appeared in its decisionmaking process. Since uncertainty issue could function in both model parameter determination and solution process, information related to decision-making becomes highly imprecise and ambiguous, especially in the current age of big data. To tackle bi-level decision problems under uncertainty, fuzzy sets and fuzzy systems have been used in terms of both decision modeling and solution approaches, formed fuzzy bi-level decision-making techniques. To help researchers understand the development experience of fuzzy bilevel decision-making techniques, and to assist practitioners to handle related decision-making problems in practice, this paper reviews the latest research on fuzzy bi-level decision-making including theoretical research results and applications. It clusters the literatures into four research directions: (1) basic fuzzy bi-level decision-making, (2) fuzzy bi-level decision-making with multiple optima, (3) fuzzy random bi-level decision-making, and (4) applications of fuzzy bi-level decision-making techniques. Several survey papers on bi-level programming/decision-making have been published in the past 20 years. However, almost the survey papers, Ben-Ayed [9] , Vicente and Calamai BIB002 , and Colson et al.   only presented overviews of bi-level decision-making, which focused on basic solution concepts and approaches, without addressing uncertain issues and fuzzy bi-level techniques. Paper by Sakawa and Nishizaki BIB005 is the only paper which reviewed fuzzy bi-level decision-making techniques. However, this paper only focused on interactive fuzzy programming approaches in terms of solving a range of stochastic bi-level and multilevel decision problems, and didn't examine the real-world application developments of fuzzy bi-level decision-making. Thus, to produce our paper is a significant work to give an overview of recent developments in fuzzy bi-level decision-making techniques. Two main types of articles reviewed in this survey: One consists of theoretical research in fuzzy bi-level decision-making; and the other consists of research on fuzzy bi-level decision-making applications including related decision support systems and case studies. The literature search and selection were performed according to the following four steps: Step 1. Publication database identification and determination: Publication databases, Science Direct, ACM Digital Library, IEEE Xplore and SpringerLink, were used to provide a comprehensive bibliography of research papers to this study. Step 2. Preliminary screening of articles: Based on related keywords of fuzzy bi-level decision-making, the articles were then selected as references if they satisfied one of the following criteria that they: (1) proposed decision models related to fuzzy bi-level decisionmaking; (2) developed solution concepts and approaches for solving fuzzy bi-level decision problems; (3) provided a decision support system tool or a realworld application in relation to fuzzy bi-level decisionmaking. Step 3. Theoretical research article selection: These papers were selected according to three criteria:  novelty -published within the last decade; (2) impactpublished in high quality (high impact factor) journals, or in conference proceedings or book chapters but with high citations; and (3) typicality -only the most typical decision models and solution approaches were retained. These papers are mainly used in Sections2-4 of this paper. Step 4. Application research article selection: Theses papers were selected according to three criteria: (1) novelty, (2) impact, and (3) coverage -reported a new or particular application domain. These papers are mainly used in Section 5. The main contributions of this paper are threefold. First, the paper comprehensively and perceptively summarizes research achievements on fuzzy bi-level decision-making techniques from the point of view of theoretical research results and applications. Second, for each fuzzy bi-level decision-making technique, it carefully analyzes typical decision models and identifies the specific requirements for solution approaches; this will directly motivate and support researchers and practitioners to promote the popularization and application of fuzzy bi-level decision-making techniques in different domains. Third, it particularly suggests several very innovative emerging research topics/directions in the area of fuzzy bi-level decisionmaking. The remainder of this paper is structured as follows. In Section 2, the techniques for basic fuzzy bi-level decision-making are reviewed. Section 3 presents technique development for fuzzy bi-level decisionmaking with multiple optima. Section 4 addresses fuzzy random bi-level decision-making techniques. Section 5 discusses the principal applications of fuzzy bi-level decision-making techniques. Our comprehensive analysis and main findings are given in Section 6.