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2016 International Conference on Mathematical, Computational and Statistical Sciences and Engineering（MCSSE 2016）

作者： Yi-fan ZHAO Shen-hua YANG Yong-feng SUO Li-yang ZHAO College of Navigation Jimei University College of Science Chongqing Normal University

In view of the linear feature of the upper-level and lower-level objective function of the bilevel programming as well as its constraint conditions, this paper presented a global optimization method. Replaced low-level sub-problem of the bilevel linear programming problem with dual problem, and then added the duality gap of the lower-level problem to the upper-level objective function, as a penalty function, and solved single level mathematical programming problem as an alternative to seek the optimal solution of the original problem. Numerical experiments show that the proposed algorithm is effective and feasible. This paper avoided the complexity with solving the dual problem's constraint domain pole, thus saved computing time and energy.

关键词： bilevel linear programming Penalization function Dual problem Duality gap Globally optimal solution

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An interval approach based on expectation optimization for fuzzy random bilevel linear programming problems

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JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY 2015年第12期66卷 2075-2085页

作者： Ren, Aihong Wang, Yuping Baoji Univ Arts & Sci Baoji Shaanxi Peoples R China Xidian Univ Xian 710071 Shaanxi Peoples R China

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 alpha-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.

关键词： bilevel linear programming fuzzy random variables interval number programming Pareto optimal solution expectation optimization

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Random fuzzy bilevel linear programming through possibility-based value at risk model

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INTERNATIONAL JOURNAL OF MACHINE LEARNING AND CYBERNETICS 2014年第2期5卷 211-224页

作者： Katagiri, Hideki Uno, Takeshi Kato, Kosuke Tsuda, Hiroshi Tsubaki, Hiroe Hiroshima Univ Fac Engn Higashihiroshima 7398527 Japan Univ Tokushima Inst Socioarts & Sci Tokushima 7708502 Japan Hiroshima Inst Technol Fac Appl Informat Sci Saeki Ku Hiroshima 7315193 Japan Doshisha Univ Fac Sci & Engn Kyotanabe 6100394 Japan Inst Stat Math Tachikawa Tokyo 1908562 Japan

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.

关键词： bilevel linear programming Random fuzzy variable Stackelberg solutions Possibility Value at risk

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Optimistic Stackelberg solutions to bilevel linear programming with fuzzy random variable coefficients

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KNOWLEDGE-BASED SYSTEMS 2014年 67卷 206-217页

作者： Ren, Aihong Wang, Yuping Xidian Univ Sch Comp Sci & Technol Xian 710071 Shaanxi Peoples R China Baoji Univ Arts & Sci Dept Math Baoji 721013 Shaanxi Peoples R China

In this paper, we consider a kind of bilevel linear programming problem where the coefficients of both objective functions are fuzzy random variables. The purpose of this paper is to develop a computational method for obtaining optimistic Stackelberg solutions to such a problem. Based on alpha-level sets of fuzzy random variables, we first transform the fuzzy random bilevel programming problem into an alpha-stochastic interval bilevel linear programming problem. To minimize the interval objective functions, the order relations which represent the decision maker's preference are defined by the right limit and the center of random interval simultaneously. Using the order relations and expectation optimization, the alpha-stochastic interval bilevel linear programming problem can be converted into a deterministic multiobjective bilevel linear programming problem. According to optimistic anticipation from the upper level decision maker, the optimistic Stackelberg solution is introduced and a computational method is also presented. Finally, several numerical examples are provided to demonstrate the feasibility of the proposed approach. (C) 2014 Elsevier B.V. All rights reserved.

关键词： bilevel linear programming Fuzzy random variables Level sets Optimistic Stackelberg solutions Interval number Multiobjective

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An interval programming approach for the bilevel linear programming problem under fuzzy random environments

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SOFT COMPUTING 2014年第5期18卷 995-1009页

作者： Ren, Aihong Wang, Yuping Xue, Xingsi Xidian Univ Sch Comp Sci & Technol Xian 710071 Peoples R China

In this paper, we address a class of bilevel linear programming problems with fuzzy random variable coefficients in objective functions. To deal with such problems, we apply an interval programming approach based on the -level set to construct a pair of bilevel mathematical programming models called the best and worst optimal models. Through expectation optimization model, the best and worst optimal problems are transformed into the deterministic problems. By means of the Kth best algorithm, we obtain the best and worst optimal solutions as well as the corresponding range of the objective function values. In this way, more information can be provided to the decision makers under fuzzy random circumstances. Finally, experiments on two examples are carried out, and the comparisons with two existing approaches are made. The results indicate the proposed approaches can get not only the best optimal solution (ideal solution) but also the worst optimal solution, and is more reasonable than the existing approaches which can only get a single solution (ideal solution).

关键词： bilevel linear programming Fuzzy random variables Interval programming Level sets Expectation optimization model

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Random fuzzy bilevel linear programming through possibility-based fractile model

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INTERNATIONAL JOURNAL OF MACHINE LEARNING AND CYBERNETICS 2014年第4期5卷 499-507页

作者： Sakawa, Masatoshi Matsui, Takeshi Hiroshima Univ Fac Engn Higashihiroshima 7398527 Japan

This paper focuses on random fuzzy noncooperative bilevel linear programming problems. Considering the probabilities that the decision makers' objective function values are smaller than or equal to target variables, fuzzy goals of the decision makers are introduced. Using the fractile model to optimize the target variables under the condition that the degrees of possibility with respect to the attained probabilities are greater than or equal to certain permissible levels, the original random fuzzy bilevel programming problems are reduced to deterministic ones. Extended concepts of Stackelberg solutions are introduced and computational methods are also presented. A numerical example is provided to illustrate the proposed method.

关键词： bilevel linear programming Random fuzzy variable Stackelberg solutions Possibility Fractile criteria optimization

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A cutting plane method for bilevel linear programming with interval coefficients

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ANNALS OF OPERATIONS RESEARCH 2014年第1期223卷 355-378页

作者： Ren, Aihong Wang, Yuping Baoji Univ Arts & Sci Dept Math Baoji 721013 Peoples R China Xidian Univ Sch Comp Sci & Technol Xian 710071 Peoples R China

This article considers the bilevel linear programming problem with interval coefficients in both objective functions. We propose a cutting plane method to solve such a problem. In order to obtain the best and worst optimal solutions, two types of cutting plane methods are developed based on the fact that the best and worst optimal solutions of this kind of problem occur at extreme points of its constraint region. The main idea of the proposed methods is to solve a sequence of linear programming problems with cutting planes that are successively introduced until the best and worst optimal solutions are found. Finally, we extend the two algorithms proposed to compute the best and worst optimal solutions of the general bilevel linear programming problem with interval coefficients in the objective functions as well as in the constraints.

关键词： bilevel linear programming Interval coefficients Cutting plane method

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bilevel linear optimization belongs to NP and admits polynomial-size KKT-based reformulations

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OPERATIONS RESEARCH LETTERS 2023年第6期51卷 618-622页

作者： Buchheim, Christoph Tech Univ Dortmund Fak Math D-44221 Dortmund Germany

It is a well-known result that bilevel linear optimization is NP-hard. In many publications, reformulations as mixed-integer linear optimization problems are proposed, which suggests that the decision version of the problem belongs to NP. However, to the best of our knowledge, a rigorous proof of membership in NP has never been published, so we close this gap by reporting a simple but not entirely trivial proof. A related question is whether a large enough "big M" for the classical KKT-based reformulation can be computed efficiently, which we answer in the affirmative. In particular, our big M has polynomial encoding length in the original problem data.(c) 2023 Elsevier B.V. All rights reserved.

关键词： bilevel linear programming KKT-reformulation NP-completeness

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Stackelberg solutions for fuzzy random bilevel linear programming through level sets and probability maximization

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OPERATIONAL RESEARCH 2012年第3期12卷 271-286页

作者： Sakawa, Masatoshi Katagiri, Hideki Matsui, Takeshi Hiroshima Univ Fac Engn Higashihiroshima 7398527 Japan

This paper considers Stackelberg solutions for bilevel linear programming problems under fuzzy random environments. To deal with the formulated fuzzy random bilevel linear programming problem, alpha-level sets of fuzzy random variables are introduced and an alpha-stochastic bilevel linear programming problem is defined for guaranteeing the degree of realization of the problem. Taking into account vagueness of judgments of decision makers, fuzzy goals are introduced and the alpha-stochastic bilevel linear programming problem is transformed into the problem to maximize the satisfaction degree for each fuzzy goal. Through probability maximization in stochastic programming, the transformed stochastic bilevel programming problem can be reduced to a deterministic bilevel programming problem. An extended concept of Stackelberg solution is introduced and a computational method is also presented. A numerical example is provided to illustrate the proposed method.

关键词： bilevel linear programming Fuzzy random variables Stackelberg solutions Level sets Probability maximization

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The Basic Properties and Algorithm of a linear bilevel programming Model with Multiple Followers

The Basic Properties and Algorithm of a Linear Bilevel Progr...

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26th Chinese Control and Decision Conference (CCDC)

作者： Deng, Shengyue Cheng, Yangjin Wang, Xinfan Hunan Univ Technol Sch Sci Zhuzhou 412007 Peoples R China

ISBN: (纸本)9781479937066

The geometric properties and optimality conditions for a class of typical model of linear bilevel programming with multiple followers are discussed in this paper. Under the assumption that the constrained set is nonempty and bounded, we study the geometric properties of the feasible set and reach the conclusion of the feasible set of weak quasi-convex and connected set. An optimum condition is obtained by using the duality theory of linear programming. According to these properties, an algorithm is proposed to solve the model and to verify the correctness of the theory and the feasibility of algorithm by giving the examples of numerical calculation.

关键词： bilevel linear programming multiple followers geometric property optimum condition

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