Fuzzy Bilevel DecisionMaking Techniques: A Survey
 Authors
 Guangquan Zhang^{a}^{, }Guangquan.Zhang@uts.edu.au, Jialin Han^{a}^{, b}^{, }Jialin.Han@student.uts.edu.au, Jie Lu^{a}^{, }Jie.Lu@uts.edu.auFaculty of Engineering and Information Technology, University of Technology Sydney, NSW 2007, AustraliaSchool of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China
 Corresponding Author
 Guangquan ZhangGuangquan.Zhang@uts.edu.au
 DOI
 https://doi.org/10.1080/18756891.2016.1180816How to use a DOI?
 Keywords
 Bilevel decisionmaking, bilevel programming, fuzzy sets, fuzzy systems, decision support systems
 Abstract
Bilevel decisionmaking techniques aim to deal with decentralized management problems that feature interactive decision entities distributed throughout a bilevel hierarchy. A challenge in handling bilevel decision problems is that various uncertainties naturally appear in decisionmaking process. Significant efforts have been devoted that fuzzy set techniques can be used to effectively deal with uncertain issues in bilevel decisionmaking, known as fuzzy bilevel decisionmaking techniques, and researchers have successfully gained experience in this area. It is thus vital that an instructive review of current trends in this area should be conducted, not only of the theoretical research but also the practical developments. This paper systematically reviews uptodate fuzzy bilevel decisionmaking techniques, including models, approaches, algorithms and systems. It also clusters related technique developments into four main categories: basic fuzzy bilevel decisionmaking, fuzzy bilevel decisionmaking with multiple optima, fuzzy random bilevel decisionmaking, and the applications of bilevel decisionmaking techniques in different domains. By providing stateoftheart knowledge, this survey paper will directly support researchers and practitioners in their understanding of developments in theoretical research results and applications in relation to fuzzy bilevel decisionmaking techniques.
 Copyright
 © 2016. the authors. Copublished by Atlantis Press and Taylor & Francis
 Open Access
 This is an open access article under the CC BYNC license (http://creativecommons.org/licences/bync/4.0/).
1. Introduction
Bilevel decisionmaking techniques, mainly implemented by bilevel programming, motivated by Stackelberg game theory [1], have been developed to address compromises between two interactive decision entities that are distributed throughout a hierarchical organization. In a bilevel decisionmaking 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 bilevel decisionmaking was proposed by Bracken and McGill [2]. A wide range of research in this area has been undertaken with the designations bilevel programming, bilevel optimization, and bilevel decisionmaking. Bilevel decisionmaking mainly is a bilevel optimization issue and mainly applies bilevel programming methods. It often appears in many decentralized management problems, and has motivated a number of researches to work on bilevel decision models [3, 4], solution approaches [5, 6] and applications [7, 8].
A challenge in handling bilevel 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 decisionmaking becomes highly imprecise and ambiguous, especially in the current age of big data. To tackle bilevel decision problems under uncertainty, fuzzy sets and fuzzy systems have been used in terms of both decision modeling and solution approaches, formed fuzzy bilevel decisionmaking techniques. To help researchers understand the development experience of fuzzy bilevel decisionmaking techniques, and to assist practitioners to handle related decisionmaking problems in practice, this paper reviews the latest research on fuzzy bilevel decisionmaking including theoretical research results and applications. It clusters the literatures into four research directions: (1) basic fuzzy bilevel decisionmaking, (2) fuzzy bilevel decisionmaking with multiple optima, (3) fuzzy random bilevel decisionmaking, and (4) applications of fuzzy bilevel decisionmaking techniques.
Several survey papers on bilevel programming/ decisionmaking have been published in the past 20 years. However, almost the survey papers, BenAyed [9], Vicente and Calamai [10], and Colson et al. [11, 12] only presented overviews of bilevel decisionmaking, which focused on basic solution concepts and approaches, without addressing uncertain issues and fuzzy bilevel techniques. Paper by Sakawa and Nishizaki [5] is the only paper which reviewed fuzzy bilevel decisionmaking techniques. However, this paper only focused on interactive fuzzy programming approaches in terms of solving a range of stochastic bilevel and multilevel decision problems, and didn’t examine the realworld application developments of fuzzy bilevel decisionmaking. Thus, to produce our paper is a significant work to give an overview of recent developments in fuzzy bilevel decisionmaking techniques.
Two main types of articles reviewed in this survey: One consists of theoretical research in fuzzy bilevel decisionmaking; and the other consists of research on fuzzy bilevel decisionmaking 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 bilevel decisionmaking, the articles were then selected as references if they satisfied one of the following criteria that they: (1) proposed decision models related to fuzzy bilevel decision making; (2) developed solution concepts and approaches for solving fuzzy bilevel decision problems; (3) provided a decision support system tool or a realworld application in relation to fuzzy bilevel decisionmaking.

Step 3. Theoretical research article selection: These papers were selected according to three criteria: (1) novelty  published within the last decade; (2) impact published 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 bilevel decisionmaking techniques from the point of view of theoretical research results and applications. Second, for each fuzzy bilevel decisionmaking 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 bilevel decisionmaking techniques in different domains. Third, it particularly suggests several very innovative emerging research topics/directions in the area of fuzzy bilevel decisionmaking.
The remainder of this paper is structured as follows. In Section 2, the techniques for basic fuzzy bilevel decisionmaking are reviewed. Section 3 presents technique development for fuzzy bilevel decisionmaking with multiple optima. Section 4 addresses fuzzy random bilevel decisionmaking techniques. Section 5 discusses the principal applications of fuzzy bilevel decisionmaking techniques. Our comprehensive analysis and main findings are given in Section 6.
2. Basic Fuzzy Bilevel DecisionMaking
This section reviews the development of techniques for solving a category of fuzzy bilevel decisionmaking 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 bilevel decision problems.
A general fuzzy linear bilevel decision problem (in the linear form) can be described as Definition 1.
Definition 1.
[13, 14] For x ∈ X ⊂ R^{p}, y ∈ Y ⊂ R^{q}, and F, f : R^{p} × R^{q} → F(R), a general fuzzy linear bilevel decision problem is defined as follows:
The original appearance of fuzzy bilevel decisionmaking was proposed by Sakawa et al. [16] on a linear bilevel 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 bilevel fractional decision problems [17]. Sakawa, et al. [18, 19] also used the interactive fuzzy programming approach and genetic algorithms to solve fuzzy bilevel 0–1 decision problems and fuzzy bilevel nonconvex decision problems. In recent years, Pramanik [20] proposed a fuzzy goal programming approach to solve fuzzy linear bilevel decision problems. Biswas and Bose [21] developed a fuzzy goal programming approach for solving quadratic bilevel 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 bilevel decision problems. However, these methods are limited to solving fuzzy bilevel 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 bilevel programming methods to solve fuzzy bilevel decision problems. Zhang and Lu [13] developed an approximation KuhnTucker approach to solve fuzzy linear bilevel decision problems. They also presented an approximation KthBest algorithm and an approximation branch and bound algorithm to solve the fuzzy linear bilevel decision problem [22, 23]. Gao et al. [24] proposed a programmable λcut set based fuzzy goal bilevel decision problem. Budnitzki [25] used the selection function approach and a modified version of the KthBest algorithm to solve a fuzzy linear bilevel decision problem. Katagiri et al. [26] defined Stackelberg solutions for fuzzy linear bilevel decision problems and then transformed fuzzy problems into deterministic linear or nonlinear bilevel decision problems that can be exactly solved by conventional bilevel solution methods. Furthermore, Zhang et al. [27] presented a fuzzy linear trilevel decision problem, and extended the linear trilevel KthBest algorithm proposed by Zhang et al. [28] to solve the problem based on λcut of a fuzzy set. Clearly, these solution methods are mainly for solving linear form of fuzzy bilevel and trilevel decisionmaking.
3. Fuzzy Bilevel DecisionMaking with Multiple Optima
In a fuzzy bilevel decision problem, multiple optima may exist at each decision level:
 (1)
The leader or follower may have multiple and conflicting objectives, and this is a fuzzy bilevel multiobjective (FBLMO) decision problem;
 (2)
Multiple decision entities may exist at the lower (follower) level, and this is known as a fuzzy bilevel multifollower (FBLMF) decision problem.
 (3)
Multiple decision entities may exist at the upper (leader) level, this is a fuzzy bilevel multileader (FBLML) decision problem.
Fuzzy bilevel decisionmaking with multiple optima has attracted numerous studies on decision models and solution methods.
In terms of research on FBLMO decisionmaking, a general linear FBLMO decision problem can be described as Definition 2.
Definition 2.
[29] For x ∈ X ⊂ R^{p}, y ∈ Y ⊂ R^{q}, and F : R^{p} × R^{q} → F^{M}(R), f : R^{p} × R^{q} → F^{N}(R), a general fuzzy linear FBLMO decision problem is formulated as follows:
Zhang et al. [29] presented the linear FBLMO decision problem as shown in Definition 2 and developed an approximation BranchandBound (BB) algorithm to solve it. Gao et al. [30] proposed a λ cut and goalprogrammingbased algorithm to solve the linear FBLMO decision problems as well. Pramanik and Dey [31] addressed a fuzzy goal programming approach for solving a linear FBLMO decision problem in which decision entities from different decision levels share the same constraints and prefer to cooperate with one another. Since the fuzzy goal programming approach does not follow the classical solution concepts of bilevel decisionmaking and neglects the upperlevel decisions, Baky et al. [32] designed an extended fuzzy goal programming algorithm for solving the FBLMO problem.
For FBLMF decisionmaking, a general linear FBLMF decision problem can be described as Definition 3 based on related research in [33, 34].
Definition 3.
[33, 34] For x ∈ X ⊂ R^{p}, y_{i} ∈ Y_{i} ⊂ R^{qi}, and F, f_{i} : R^{p} × R^{q1} ×…× R^{qk} → F(R), i = 1,2,…k, a general linear FBLMF decision problem in which one leader and k followers are involved can be written as:
Gao et al. [33] focused on problems of fuzzy linear bilevel decisionmaking with multiple followers who share a common objective but have different constraints. The authors developed a partial swarm optimization (PSO) algorithm for solving these problem based on the ranking relationship among fuzzy sets defined by cutset and satisfactory degree. Gao et al. [34] also studied a linear FBLMF decision problem in which multiple followers share the common constraints and developed a PSO algorithm to solve the problem. Gao and Liu [35] integrated fuzzy simulation, neural network and genetic algorithm to produce a hybrid intelligent algorithm for solving a nonlinear FBLMF problem.
In order to solve fuzzy bilevel multiobjective multifollower (FBLMOMF) decision problems, Zhang et al. [36] proposed a set of linear models and developed an extended BB algorithm. Based on it, Zhang et al. [37] developed an approximation BB algorithm to solve a linear FBLMOMF decision problem with a partial cooperative relationship between multiple followers. Zhang et al. [38] also proposed a set of decision models to describe complex FBLMOMF decision issues and presented an approximation BB algorithm to solve a linear FBLMOMF decision problem in which focusing followers share the same decision variables. Zhang and Lu [39] presented an approximation KthBest algorithm to solve a linear FBLMOMF decision problem with a cooperative relationship between multiple followers. These developments provide a strong foundation for fuzzy bilevel decisionmaking techniques in various situations.
4. Fuzzy Random Bilevel DecisionMaking
Fuzziness and randomness are two main sources of uncertainty [40]. Bilevel decisionmaking may appear in not only a fuzzy situation but also a random environment, and this is known as fuzzy random bilevel decisionmaking. Related parameters in relation to fuzzy random bilevel decisionmaking 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 [41]. With respect to Definition 1, if the coefficients of decision variables are characterized by fuzzy random variables, the fuzzy bilevel program (1) is known as a fuzzy random bilevel decision problem.
In recent years, attention has been increasingly paid to fuzzy random bilevel decisionmaking. Sakawa et al. [42] considered a linear bilevel decision problem under a fuzzy random environment and transformed the problem into a deterministic bilevel decision problem through level sets and probability maximization for finding a Stackelberg solution. Sakawa et al. [43] also used another different method based on expectation optimization using possibility and necessity to reduce a fuzzy random linear bilevel decision problem to a deterministic bilevel decision problem for solving. Ren and Wang [44] transformed a fuzzy random linear bilevel decision problem into a deterministic bilevel multiobjective decision problem using level sets and expectation optimization, and found a optimistic Stackelberg solution to the considered problem. Ren et al. [45] and Ren and Wang [46] developed interval programming approaches for solving fuzzy random linear bilevel decision problems. Katagiri et al. [47] introduced the concept of value at risk and possibility theory, and transformed a fuzzy random bilevel decision problem into a set of deterministic problems that can be solved by nonlinear bilevel decisionmaking techniques.
5. Applications of Fuzzy Bilevel DecisionMaking Techniques
Fuzzy bilevel decisionmaking techniques have been widely applied to handle decentralized management problems in the real world that often appear in imprecise and ambiguous environments. These applications largely fall into the following four domains: (1) supply chain and logistics management; (2) planning and scheduling operations; (3) resource allocation in hierarchy; and (4) energy management and environment protection.
5.1. Supply chain and logistics management
In terms of applications in supply chain and logistics management, Gao et al. [48] established a fuzzy bilevel decision model for buyers and vendors in a pricing supply chain that some coefficients are generally obtained from experiments and cannot be defined as crisp numbers. A PSO algorithm was developed to solve the resulting model. Sun and Gao [49] proposed a bilevel decision model to handle pricing and production decisions with fuzzy demand and fuzzy cost parameters. The upper level (leader) aims to determine the optimal price and production quantity with capacity constraints, whereas the lower level (follower) problem tries to structure the distribution pattern of customers with the aim of satisfying the demand at minimum cost. Ji and Shao [50] addressed a bilevel decision model for a newsboy problem with fuzzy demands and quantity discounts in a hierarchical decision system, in which manufacturer at the upper level decides the wholesale prices of newspaper in order to maximize its profit, while retailers at the lower level decide ordering quantities of newspaper in order to maximize their own expected profits. Zhang et al. [51] applied fuzzy bilevel optimization techniques to model and support bidding decisionmaking between the generating company and the market operator in electricity markets. Lan et al. [52] studied an optimal taxation problem with asymmetric information in which the government and the monopolist are principals while the consumer is their agent. The authors formulated this problem as a fuzzy nonlinear bilevel decision model with the purpose of maximizing the expected social welfare and the monopolist’s expected welfare under an incentive feasible mechanism.
5.2. Planning and scheduling operations
For planning and scheduling operations in a decentralized manner, Abass [53] presented a fuzzy bilevel programming approach to solve a flow shop scheduling problem with fuzzy processing times in a hierarchy of two decision entities, in which the shop owner at the higher level assigns jobs to machines in order to minimize the flow time, while the customer at the lower level decides on a job schedule in order to minimize the makespan. Konur and Golias [54] studied scheduling of inbound trucks at inbound doors of a crossdock facility under truck arrival time uncertainty. The authors formulated this problem as pessimistic and optimistic bilevel decision problems respectively in which the uncertain truck arrival time is handled by fuzzy number with a triangular distribution. Also, the authors developed a genetic algorithm to solve the bilevel pessimistic and optimistic formulations.
5.3. Resource allocation in hierarchy
In regard to handling resource allocation in hierarchy, Gang et al. [55] focused on a multiproject resource allocation problem using bilevel optimization. In this problem, the company manager at the upper level aims to allocate the company’s resources to multiple projects to achieve the lowest cost that include resource costs and a tardiness penalty, while each project manager at the lower level attempts to schedule their resourceconstrained project with the minimization of project duration. The authors applied a fuzzy random bilevel decision model to describe this problem in which fuzzy random variables are used to model uncertain activity durations and resource costs. Xu and Gang [56] studied a material flow traffic assignment problem with fuzzy random variables in a largescale construction project. A fuzzy random bilevel multiobjective decision model was established in which the transportation time and cost in each arc are considered as fuzzy random variables, and the authors developed a PSO algorithm to solve the resulting model.
5.4. Energy management and environment protection
Fuzzy bilevel decisionmaking techniques have been also applied to handle energy management problems in relation to the consumption of resources and the generation of waste. Aviso et al. [57] developed a fuzzy bilevel optimization model to explore the effect of charging fees for the purchase of freshwater and the treatment of wastewater in optimizing the water exchange network of plants in an ecoindustrial park (EIP). Tan et al. [58] extended the optimization model developed by Aviso et al. [57] to a new fuzzy bilevel decision model by modifying the role of the EIP authority to include water regeneration and redistribution via a centralized hub, and found a reasonable compromise between the EIP authority’s desire to minimize fresh water usage and the participating companies’ desire to minimize costs. Xu et al. [59] developed a fuzzy random bilevel decision model for tackling a regional water resources allocation problem on the basis of water rights distribution in a river basin. The bilevel decision model targets to optimize the total benefit of the society and the economic benefit of each subarea respectively for the upper and lower levels.
From the aspect of environment protection and risk analysis, Gan and Xu [60] considered a problem of hedging against seismic risk through the retrofit of transportation systems in largescale construction projects (LSCP). They formulated this problem as a fuzzy random multiobjective bilevel decision model with the objectives of the retrofit costs and the benefits on two separate levels. Xu et al. [61] focused on a bilevel decision problem of hazmat transportation network design, which considers an location problem of emergency response teams. The authority at the upper level designs the transportation network to minimize the total transportation risk, while the carriers at the lower level choose their routes with the aim of minimizing the total transportation cost. The problem was formulated as fuzzy bilevel optimization model in which a complex fuzzy variable was used to model transportation risk, and then an improved artificial bee colony algorithm with prioritybased encoding was applied to find an optimal solution to the resulting model. For the sake of optimizing regional economies, social employment, and water quality protection, Xu et al. [62] proposed a fuzzy random multiobjective bilevel optimization model for constructed wetland planning in which fuzzy random variables are used to describe uncertainties in the system. Yao et al. [63] gave a fuzzy bilevel multiobjective optimization model to handle a stone resource assignment problem with the aim of decreasing dust and waste water emissions. A fuzzy simulationbased improved simulated annealing (FSISA) algorithm was developed to find the Pareto optimal solutions.
In addition to these fuzzy bilevel decisionmaking applications in a range of domains, related decision support systems (software) have been developed to assist decision markers in solving fuzzy bilevel decision problems. Gao et al. [24] developed a fuzzy bilevel decision support system software based on fuzzy goal bilevel decisionmaking techniques, as shown in Fig. 1, and applied the software to handle a problem of traffic management and risk analysis in a disaster area. The authors also applied the fuzzy bilevel decision support system to deal with a manufacturerretailer decision problem in an uncertain marketing situation [64].
6. Comprehensive Analysis and Prospective Research Topics
In this paper, we have reported the new developments in fuzzy bilevel decisionmaking techniques including basic fuzzy bilevel decisionmaking, fuzzy bilevel decisionmaking with multiple optima, fuzzy random bilevel decisionmaking, and their applications in handling realworld cases through a comprehensive review. This review demonstrates the following features of fuzzy bilevel decisionmaking study:
 (1)
A number of solution approaches/algorithms have been developed to solve a range of fuzzy bilevel decision problems. The majority of the solution approaches are based on the idea that transforms a fuzzy bilevel decision problem into a deterministic decision problem for solving in the first place.
 (2)
Motivated by the constant complexity of uncertain situations, an increasing number of papers are devoted to fuzzy bilevel decisionmaking with multiple optima and fuzzy random bilevel decisionmaking. A range of advanced fuzzy bilevel decision models and effective solution algorithms have been developed to handle these complex situations.
 (3)
Fuzzy bilevel decisionmaking techniques have been widely applied to handle decentralized management and optimization problems under uncertainty in various domains. The application domains and fuzzy bilevel decisionmaking techniques used in applications are summarized in Table 1, which shows the number of each category of fuzzy bilevel decisionmaking techniques applied in each domain.
Application domains  Decisionmaking techniques  

Basic fuzzy bilevel decisionmaking  Fuzzy bilevel decisionmaking with multiple optima  Fuzzy random bilevel decisionmaking  No. of listed references  
Supply chain and logistics management  4  4  
Planning and scheduling operations  2  2  
Resource allocation in hierarchy  2  2  
Energy management and environment protection  4  2  3  9 
Total  10  2  5  17 
Summary of fuzzy bilevel decisionmaking techniques in each application domain
Although fuzzy bilevel decisionmaking techniques have attracted much attention, several evident challenges still require further research:
 (1)
Computational complexity is a crucial issue in solving fuzzy bilevel decision problems. Although many solution algorithms have been developed, it is necessary and urgent to develop much more efficient algorithms, e.g. heuristic algorithms, for solving much more general formulations, e.g. nonlinear problems with complex membership functions, and solving largescale versions in practically reasonable computational time. Moreover, fully fuzzy bilevel decision problems, in which both coefficients and variables are characterized by fuzzy numbers, are also an emerging research topic with respect to computational complexity and need to be explored in depth.
 (2)
Bilevel decision problems nowadays often appear in a highly uncertain decision environment, thus, this requires further research on how to wisely model such uncertain problems by means of a fuzzy bilevel decision support system. Many bilevel decision problems under fuzziness in the real world cannot be modeled as mathematical programming formulations. Consequently, it is a challenge to break through mathematical programs and integrate the principle of fuzzy bilevel decisionmaking with other decisionmaking techniques to handle a much wider range of decision problems under uncertainty.
 (3)
A new direction in handling bilevel decisionmaking is datadriven modeling in the current age of big data. How to implement datadriven bilevel decisionmaking under uncertainty using data analytical techniques and cognitive decisionmaking methods [65] can be considered as a future research issue.
Two important features of the paper clearly distinguish it from previous survey papers in the bilevel/multilevel decisionmaking area. On the one hand, in contrast to previous survey papers that only review deterministic bilevel decisionmaking techniques, this survey targets the latest bilevel decisionmaking using fuzzy techniques and their comprehensive development. On the other hand, apart from theoretical research developments, this survey systematically examines the realworld application developments of fuzzy bilevel decisionmaking techniques in different domains. We believe that this paper provides researchers and practitioners with the stateoftheart knowledge on the development of fuzzy bilevel decisionmaking approaches and gives guidelines about how to apply fuzzy bilevel decisionmaking techniques to deal with various decision activities under uncertainty in realworld applications.
Acknowledgements
This work is supported by the Australian Research Council (ARC) under discovery grant DP140101366.
References
Cite this article
TY  JOUR AU  Guangquan Zhang AU  Jialin Han AU  Jie Lu PY  2017 DA  2017/01 TI  Fuzzy Bilevel DecisionMaking Techniques: A Survey JO  International Journal of Computational Intelligence Systems SP  25 EP  34 VL  9 IS  sup1 SN  18756883 UR  https://doi.org/10.1080/18756891.2016.1180816 DO  https://doi.org/10.1080/18756891.2016.1180816 ID  Zhang2017 ER 