Fuzzy Bi-level Decision-Making Techniques: A Survey
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paper_title: On bilevel multi-follower decision making: General framework and solutions

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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.
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paper_title: Multifollower Trilevel Decision Making Models and System

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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.
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paper_title: Mathematical Programs with Optimization Problems in the Constraints

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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)
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paper_title: Interactive fuzzy programming for multi-level programming problems: a review

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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.
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paper_title: Bilevel and Multilevel Programming: A Bibliography Review

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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.
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paper_title: Model and approach of fuzzy bilevel decision making for logistics planning problem

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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 ...
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paper_title: Practical Bilevel Optimization: Algorithms and Applications

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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.
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paper_title: The Definition of Optimal Solution and an Extended Kuhn-Tucker Approach for Fuzzy Linear Bilevel Programming

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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.
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paper_title: The solution approach to linear fuzzy bilevel optimization problems

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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.
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paper_title: A fuzzy programming approach for solving quadratic bilevel programming problems with fuzzy resource constraints

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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.
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paper_title: A fuzzy tri-level decision making algorithm and its application in supply chain

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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.
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paper_title: Possibilistic Stackelberg solutions to bilevel linear programming problems with fuzzy parameters

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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.
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paper_title: Practical Bilevel Optimization: Algorithms and Applications

paper_content:

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.
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paper_title: FUZZY BILEVEL PROGRAMMING: MULTI-OBJECTIVE AND MULTI-FOLLOWER WITH SHARED VARIABLES

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Bilevel programming deals with hierarchical optimization problems in which the leader at the upper level attempts to optimize his or her objectives, but subject to a set of constraints and the follower's reactions. Typical bilevel programming considers one leader one follower situation and supposes each of them has only one objective. In real world situations, multiple followers may be involved and they may be with different relationships such as sharing decision variables or not, sharing objectives or not. Therefore, the leader's decision will be affected not only by those followers' reactions but also by their relationships. In addition, any of the leader and/or these followers may have multiple conflict objectives that should be optimized simultaneously. Furthermore, the parameters of a bilevel programming model may be described by uncertain values. This paper addresses all these three issues as a whole by particularly focusing on the situation of sharing decision variables among followers. It first proposes a set of fuzzy multi-objective multi-follower bilevel programming (FMMBP) models to describe the complex issue. It then presents an approximation branch-and-bound algorithm to solve the FMMBP problems. Finally, two examples illustrate the proposed models and algorithm.
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paper_title: Fuzzy bilevel programming with multiple objectives and cooperative multiple followers

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Classic bilevel programming deals with two level hierarchical optimization problems in which the leader attempts to optimize his/her objective, subject to a set of constraints and his/her follower's solution. In modelling a real-world bilevel decision problem, some uncertain coefficients often appear in the objective functions and/or constraints of the leader and/or the follower. Also, the leader and the follower may have multiple conflicting objectives that should be optimized simultaneously. Furthermore, multiple followers may be involved in a decision problem and work cooperatively according to each of the possible decisions made by the leader, but with different objectives and/or constraints. Following our previous work, this study proposes a set of models to describe such fuzzy multi-objective, multi-follower (cooperative) bilevel programming problems. We then develop an approximation Kth-best algorithm to solve the problems.
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paper_title: A Particle Swarm Optimization Based Algorithm for Fuzzy Bilevel Decision Making with Objective-Shared Followers

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A bilevel decision problem may have multiple followers as the lower decision units and have fuzzy demands simultaneously. This paper focuses on problems of fuzzy linear bilevel decision making with multiple followers who share a common objective but have different constraints (FBOSF). Based on the ranking relationship among fuzzy sets defined by cut set and satisfactory degree, a FBOSF model is presented and a particle swarm optimization based algorithm is developed.
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paper_title: Bi-level multi-objective programming problem with fuzzy demands: a fuzzy goal programming algorithm

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In this paper, a fuzzy goal programming (FGP) algorithm for solving bi-level multi-objective programming problems with fuzzy demands is presented. These fuzzy demands reflect the experts' imprecise or fuzzy understandings of the nature of parameters in the problem formulation process are assumed to be characterized as fuzzy numbers. Using the level sets of fuzzy parameters, the corresponding non fuzzy bi-level programming problem is introduced. In the proposed algorithm, the membership functions for the defined fuzzy goals of all objective functions at the two levels, as well as the membership functions for the vector of decision variables controlled by FLDM are developed in the model formulation of the problem. Then FGP algorithm is used to achieve the highest degree of each of the membership goals by minimizing their deviational variables and thereby obtaining the most satisfactory solution for all decision makers. Illustrative numerical example is given to demonstrate the proposed algorithm.
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paper_title: Bi-level multi- objective programming problem with fuzzy parameters

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paper deals with fuzzy goal programming approach to bi- level multi-objective programming problem with fuzzy parameters. In the proposed approach, the tolerance membership functions for the fuzzily described objective functions are defined by determining individual optimal solution of the objective functions of each of the decision makers. Since the objectives are potentially conflicting in general, possible relaxations of both level decisions are considered by providing preference bounds on the decision variables for avoiding decision deadlock. Then fuzzy goal programming technique is used for achieving highest degree of each of the membership goals by minimizing negative deviational variables. An algorithm is presented with termination criteria. An illustrative numerical example is provided to demonstrate the efficiency of the proposed approach.
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paper_title: Models and Algorithm for Fuzzy Multi-objective Multi-follower Linear Bilevel Programming

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Basic bilevel programming deals with hierarchical optimization problems in which the leader at the upper level attempts to optimize his/her objective, subject to a set of constraints and his/her follower's solution, and the follower at the lower level tries to find an optimized strategy according to each of possible decisions made by the leader. Three issues may be involved in a basic bilevel decision problem. One is that bilevel decision making model may involve uncertain parameters which appear either in the objective functions or constraints of the leader or the follower or both. Second, the leader and the follower may have multiple conflict objectives that should be optimized simultaneously. Third, there may have multiple followers in a real decision situation. Following our previous work, this study proposes a set of fuzzy multi-objective multi-follower linear bilevel programming models to describe the three issues. It also develops an approximation branch-and-bound algorithm to solve such kinds of problems.
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paper_title: Fuzzy multilevel programming with a hybrid intelligent algorithm

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In order to model fuzzy decentralized decision-making problem, fuzzy expected value multilevel programming and chance-constrained multilevel programming are introduced. Furthermore, fuzzy simulation, neural network, and genetic algorithm are integrated to produce a hybrid intelligent algorithm for finding the Stackelberg-Nash equilibrium. Finally, two numerical examples are provided to illustrate the effectiveness of the hybrid intelligent algorithm.
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paper_title: An algorithm for fuzzy multi-objective multi-follower partial cooperative bilevel programming

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In a bilevel decision problem, both the leader and the follower may have multiple objectives to optimize under certain constraints. In the meantime, these objective functions and constraints may contain some uncertain parameters. In addition, there may have multiple followers involved in a bilevel decision problem. These followers may share their individual decision variables with each other but keep individual objectives in reacting any of the leader's decisions, which is a common situation in real bilevel decision activities. This study deals with all above three issues, fuzzy parameters, multi-objectives, and multi-followers in a partial cooperative situation, at the same time. After a set of models for describing different cases of the fuzzy multi-objective multi-follower bilevel programming with partial cooperation (FMMBP-PC) problem, this paper develops an approximation branch-and-bound algorithm to solve this problem.
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paper_title: A particle swarm optimization based algorithm for fuzzy bilevel decision making with constraints-shared followers

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In a bilevel decision problem, decision making may involve multiple followers and fuzzy demands. This research focuses on the problem of fuzzy linear bilevel decision making with multiple followers who share common constraints (FBCSF). Based on the ranking relationship among fuzzy sets defined by cut set and satisfactory degree α, a FBCSF model is presented and a particle swarm optimization based algorithm is developed. The experiments reveal that solutions obtained by this algorithm are reasonable and stable.
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paper_title: Random fuzzy bilevel linear programming through possibility-based value at risk model

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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.
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paper_title: Fuzzy random variables

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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.
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paper_title: Fuzzy random bilevel linear programming through expectation optimization using possibility and necessity

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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.
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paper_title: An Interval Approach Based on Expectation Optimization for Fuzzy Random Bilevel Linear Programming Problems

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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.
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paper_title: Fuzzy random variables—I. definitions and theorems

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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.
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paper_title: Model and algorithm for bilevel newsboy problem with fuzzy demands and discounts

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This paper considers the model for the newsboy problem with fuzzy demands and quantity discounts in hierarchical decision system, in which the manufacturer is considered as the decision-maker (DM) of the top level and decides the wholesale prices of newspaper in order to maximize his profit, while the retailers are the DMs of the lower level and decide the ordering quantities of newspaper in order to maximize their own expected profits. A bilevel programming model is formulated for this kind of problem. To solve the model, a hybrid intelligent algorithm based on genetic algorithm and fuzzy simulation is developed, and some numerical examples are given to illustrate its effectiveness.
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paper_title: A Fuzzy Bi-level Pricing Model and a PSO Based Algorithm in Supply Chains

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Due to rapid technological innovation and severe competition, the upstream component price and the downstream product cost in hi-tech industries usually decline significantly with time. In building a pricing supply chain model, some coefficients are generally obtained from experiments and cannot be defined as crisp numbers. Thus, an effective fuzzy pricing supply chain model becomes crucial. This paper establishes a fuzzy bi-level pricing model for buyers and vendors in supply chains. Then, a particle swarm optimization (PSO) based algorithm is developed to solve problems defined by this model. Experiments show that this PSO-based algorithm can solve fuzzy bi-level pricing problems effectively.
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paper_title: Bilevel Model for Pricing and Production Planning Decision with Fuzzy Parameters

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With the deepen of market competition, product pricing and production decision problem in many firms have become more and more important. A bilevel model is proposed to describe the pricing and production decisions with fuzzy demand and fuzzy cost parameters. The upper level is to determine the optimal price and production quantity with capacity constraints. Using this information, the lower level problem tries to structure a response (the distribution pattern of customers (or markets)) that will satisfy his demand at minimum cost. And after transforming the fuzzy numbers into the crisp value by Graded Mean Integration Representation method, the solution algorithm based on difference method is given. Finally, the application of the model and its algorithm are illustrated with a simple example.
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paper_title: Bilevel programming approach applied to the flow shop scheduling problem under fuzziness

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This paper presents a fuzzy bilevel programming approach to solve the flow shop scheduling problem. The problem considered here differs from the standard form in that operators are assigned to the machines and imposing a hierarchy of two decision makers with fuzzy processing times. The shop owner considered higher level and assigns the jobs to the machines in order to minimize the flow time while the customer is the lower level and decides on a job schedule in order to minimize the makespan. In this paper, we use the concepts of tolerance membership function at each level to define a fuzzy decision model for generating optimal (satisfactory) solution for bilevel flow shop scheduling problem. A solution algorithm for solving this problem is given. Copyright Springer-Verlag Berlin/Heidelberg 2005
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paper_title: Analysis of different approaches to cross-dock truck scheduling with truck arrival time uncertainty

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This paper studies scheduling of inbound trucks at the inbound doors of a cross-dock facility under truck arrival time uncertainty. Arrival time of an inbound truck is considered to be unknown. In particular, the cross-dock operator only acknowledges the arrival time window of each truck, i.e., the lower and upper bounds of any inbound truck's arrival time. In absence of any additional information, the cross-dock operator may use three approaches to determine a scheduling strategy: deterministic approach (which assumes expected truck arrival times are equal to their mid-arrival time windows), pessimistic approach (which assumes the worst truck arrivals will be realized), and optimistic approach (which assumes the best truck arrivals will be realized). In this paper, a bi-level optimization problem is formulated for pessimistic and optimistic approaches. We discuss a Genetic Algorithm (GA) to solve the truck-to-door assignments for given truck arrival times, which solves the deterministic approach. Then the GA is modified to solve the bi-level formulations of the pessimistic and the optimistic approaches. Our numerical studies show that an hybrid approach regarding the pessimistic and the optimistic approaches may outperform all of the three approaches in certain cases.
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paper_title: Multiproject Resources Allocation Model under Fuzzy Random Environment and Its Application to Industrial Equipment Installation Engineering

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This paper focuses on a multiproject resource allocation problem in ::: a bilevel organization. To solve this problem, a bilevel multiproject resource allocation model under a fuzzy random ::: environment is proposed. Two levels of decision makers are ::: considered in the model. On the upper level, the company manager ::: aims to allocate the company's resources to multiple projects to ::: achieve the lowest cost, which include resource costs and a ::: tardiness penalty. On the lower level, each project manager attempts ::: to schedule their resource-constrained project, with minimization of ::: project duration as the main objective. In contrast to prior ::: studies, uncertainty in resource allocation has been explicitly ::: considered. Specifically, our research uses fuzzy random variables ::: to model uncertain activity durations and resource costs. To search ::: for the optimal solution of the bilevel model, a hybrid algorithm ::: made up of an adaptive particle swarm optimization, an adaptive ::: hybrid genetic algorithm, and a fuzzy random simulation algorithm is ::: also proposed. Finally, the efficiency of the proposed model and ::: algorithm is evaluated through a practical case from an industrial ::: equipment installation company. The results show that the proposed ::: model is efficient in dealing with practical resource allocation problems in a bilevel organization.
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paper_title: Multi-objective bilevel construction material transportation scheduling in large-scale construction projects under a fuzzy random environment

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This paper investigates a transportation scheduling problem in large-scale construction projects under a fuzzy random environment. The problem is formulated as a fuzzy, random multi-objective bilevel optimization model where the construction company decides the transportation quantities from every source to every destination according to the criterion of minimizing total transportation cost and transportation time on the upper level, while the transportation agencies choose their transportation routes such that the total travel cost is minimized on the lower level. Specifically, we model both travel time and travel cost as triangular fuzzy random variables. Then the multi-objective bilevel adaptive particle swarm optimization algorithm is proposed to solve the model. Finally, a case study of transportation scheduling for the Shuibuya Hydropower Project in China is used as a real world example to demonstrate the practicality and efficiency of the optimization model and algorithm.
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paper_title: Bi-level fuzzy optimization approach for water exchange in eco-industrial parks

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In order to minimize the consumption of resources and the generation of waste, eco-industrial parks (EIPs) have been designed to encourage the establishment of waste exchange networks between the plants contained within them. Considering that the participating plants have their individual fuzzy cost goals and that the park authority has the objective of minimizing total freshwater consumption in the EIP, this problem may be formulated as a bi-level optimization problem, with the park authority as the upper-level decision-maker. A bi-level fuzzy optimization model is developed 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 EIP. The effect of subsidizing exchange of streams between plants is also included in the model. Results show that it is not necessary to impose exorbitant fees, nor to completely subsidize costs associated with recycling, in order to induce significant reduction of the consumption of freshwater in the EIP.
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paper_title: Constructed Wetland Planning-Based Bilevel Optimization Model under Fuzzy Random Environment: Case Study of Chaohu Lake

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AbstractTo optimize regional economies, social employment, and water quality protection, a multiobjective bilevel optimization model under a fuzzy random environment based on constructed wetland planning is developed in this study. Fuzzy random variables are used to describe the uncertainties in the system, and three special fuzzy random simulation methods are proposed to address the fuzzy random variable calculations. From the inherent bilevel model interaction, a new algorithm called the fuzzy random simulation-based nested genetic algorithm (FRS-based NGA) is designed as an intelligent solution to solve the model. Then, the model is applied to a real-world case: the Chaohu Lake watershed, China. The results under different objective values, probability levels, and possibility levels are compared so that an optimal scheme for regional economic development, social employment, and the wetland construction scale can be identified. Finally, the comparative analysis, sensitivity analysis, and convergence ana...
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paper_title: A Fuzzy Multi-Objective bilevel Decision Support System

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In a bilevel decision problem, both the leader and the follower may have multiple objectives, and the coefficients involved in these objective functions or constraints may be described by some uncertain values. To express such a situation, a fuzzy multi-objective bilevel (FMOBL) programming model and related solution methods are introduced. This research develops a FMOBL decision support system through implementing the proposed FMOBL methods.
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paper_title: Bilevel Optimization of Regional Water Resources Allocation Problem under Fuzzy Random Environment

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AbstractThe allocation of water resources has long been recognized as a critical optimization problem. In this study, a bilevel programming model with fuzzy random variables is developed for tackling a regional water resources allocation problem on the basis of water rights distribution in a river basin. The bilevel programming model takes the optimal total benefit of the society and the optimal economic benefit of each subarea as the upper and lower targets, respectively. In contrast to previous studies, the balance of the satisfactory degree between the upper and lower decision makers is considered in the bilevel optimization for ensuring the equity of the water resource allocation. To deal with inherent uncertainties, the fuzzy random variables are first transformed into trapezoidal fuzzy numbers, and by taking advantage of the expected value operation, the trapezoidal fuzzy numbers are subsequently defuzzified. For solving the complex and nonlinear bilevel programming model, an interactive fuzzy progr...
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paper_title: Hazmats Transportation Network Design Model with Emergency Response under Complex Fuzzy Environment

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A bilevel optimization model for a hazardous materials transportation network design is presented which considers an emergency response teams location problem. On the upper level, the authority designs the transportation network to minimize total transportation risk. On the lower level, the carriers first choose their routes so that the total transportation cost is minimized. Then, the emergency response department locates their emergency service units so as to maximize the total weighted arc length covered. In contrast to prior studies, the uncertainty associated with transportation risk has been explicitly considered in the objective function of our mathematical model. Specifically, our research uses a complex fuzzy variable to model transportation risk. An improved artificial bee colony algorithm with priority-based encoding is also applied to search for the optimal solution to the bilevel model. Finally, the efficiency of the proposed model and algorithm is evaluated using a practical case and various computing attributes.
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paper_title: A note on an extended fuzzy bi-level optimization approach for water exchange in eco-industrial parks with hub topology

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Abstract In our previous paper, a fuzzy bi-level programming model was developed to determine optimal interplant water integration networks in eco-industrial parks (EIPs). This approach allowed the appropriate incentive mechanisms, in the form of fresh water and effluent fees as well as water reuse subsidies, to be optimized from the perspective of the EIP authority. This work extends the original mathematical model by modifying the role of the EIP authority to include water regeneration and redistribution via a centralized hub. The resulting fuzzy bi-level programming model may then be solved to yield a “satisficing” solution that reflects a reasonable compromise between the EIP authority's desire to minimize fresh water usage, and the participating companies’ desire to minimize costs. A case study is used to illustrate the modeling approach.
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paper_title: Retrofitting Transportation Network Using a Fuzzy Random Multiobjective Bilevel Model to Hedge against Seismic Risk

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This paper focuses on the problem of hedging against seismic risk through the retrofit of transportation systems in large-scale construction projects (LSCP). A fuzzy random multiobjective bilevel programming model is formulated with the objectives of the retrofit costs and the benefits on two separate levels. After establishing the model, a fuzzy random variable transformation approach and fuzzy variable approximation decomposition are used to deal with the uncertainty. An approximation decomposition-based multi-objective AGLNPSO is developed to solve the model. The results of a case study validate the efficiency of the proposed approach.
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paper_title: A Stone Resource Assignment Model under the Fuzzy Environment

paper_content:

This paper proposes a bilevel multiobjective optimization model with fuzzy coefficients ::: to tackle a stone resource assignment problem with the aim of decreasing dust and waste water emissions. On the upper level, the local government wants to assign a reasonable exploitation amount to each stone plant so as to minimize total emissions and maximize employment and economic profit. On the lower level, stone plants must reasonably assign stone resources to produce different stone products under the exploitation constraint. To deal with inherent uncertainties, the object functions and constraints are defuzzified using a possibility measure. A fuzzy simulation-based improved simulated annealing algorithm (FS-ISA) is designed to search for the Pareto optimal solutions. Finally, a case study is presented to demonstrate the practicality and efficiency of the model. Results and a comparison analysis are presented to highlight the performance of the optimization method, which proves to be very efficient compared with other algorithms.
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Title: Fuzzy Bi-level Decision-Making Techniques: A Survey
Section 1: Introduction
Description 1: Provide an overview of bi-level decision-making including historical context and key concepts. Highlight the motivation behind using fuzzy bi-level decision-making techniques and state the main contributions and structure of the survey.

Section 2: Basic Fuzzy Bi-level Decision-Making
Description 2: Review techniques for solving basic fuzzy bi-level decision-making problems where each decision level optimizes a unique objective. Discuss various solution methods and their applications.

Section 3: Fuzzy Bi-level Decision-Making with Multiple Optima
Description 3: Discuss the complexities and methodologies for handling fuzzy bi-level decision-making problems with multiple and potentially conflicting optima at different decision levels. Cover different decision model types such as multi-objective, multi-follower, and multi-leader.

Section 4: Fuzzy Random Bi-level Decision-Making
Description 4: Explore decision-making under combined fuzziness and randomness, known as fuzzy random bi-level decision-making. Elaborate on the unique challenges and approaches for modeling and solving these problems.

Section 5: Applications of Fuzzy Bi-level Decision-Making Techniques
Description 5: Highlight the real-world applications of fuzzy bi-level decision-making techniques in various domains such as supply chain management, planning and scheduling operations, resource allocation, and energy management. Provide example case studies and developed decision support systems.

Section 6: Comprehensive Analysis and Prospective Research Topics
Description 6: Summarize key findings from the review and identify existing challenges and gaps in current methodologies. Suggest prospective research topics and emerging trends in fuzzy bi-level decision-making for future exploration.