模糊随机多目标决策模型及其在资产组合选择中的应用
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摘要
在现实世界中，人们制定决策时经常会碰到各种不确定现象，其中随机现象和模糊现象是两种主要的不确定现象．随机性是指事件是否发生的不确定性，用来描述和刻画随机现象的工具是随机变量．模糊性是指事件本身状态的不确定性，用来描述和刻画模糊现象的根据是模糊集．随机多目标规划和模糊多目标规划可以帮助人们分别在随机不确定环境和模糊不确定环境下做出决策．然而，在实际决策过程中，人们面临的常常是双重不确定性环境，即随机现象和模糊信息同时存在并相互融合，无法截然分开．模糊随机变量是双重不确定变量的一种，它是描述双重不确定性现象(指模糊随机现象)的一种有用的数学工具．模糊随机变量被定义为从概率空间映射到模糊变量构成的集合上的可测函数．简单的说，模糊随机变量就是一个取值为模糊集的随机变量．模糊随机现象在现实生活中广泛存在．关于模糊随机多目标决策问题的研究不但具有理论意义，同时也具有实际应用意义．
资产组合选择理论研究如何把投资资金分配到不同的资产中，以达到分散风险并确保收益的目的．在证券市场中存在着多种不确定性，既有随机性也有模糊性．到目前为止，多数资产组合选择模型都是以股票的未来收益率是随机变量为前提并建立在概率论基础上的．最近，证券市场中存在的模糊不确定性逐渐被人们认识到，在假设股票的未来收益率为模糊变量的条件下，各种模糊资产组合选择模型被相继提出．模糊资产组合选择问题的研究成为一个新的极有前途的研究方向．实际上，无论是随机或模糊资产组合选择模型，它们往往只考虑了一种不确定性，对在双重不确定环境下的资产组合选择问题需要做进一步的研究．
为此，本文在广泛借鉴和吸收国内外研究成果的基础上，以模糊随机理论为基础，对模糊随机多目标规划模型和算法进行了研究．同时，利用模糊随机变量刻画证券市场中存在的模糊随机现象，建立了若干在模糊随机环境下的资产组合选择模型(其中包括模糊随机环境下的多目标资产组合选择模型)．主要工作如下：
(1)为刻画证券市场中同时出现的随机不确定性和模糊不确定性，本文将股票的未来收益率处理为模糊随机变量，同时考虑了历史交易数据，专家的经验和知识以及投资者个人对各股票未来收益的预期三方面因素．
(2)遵循Markowitz提出的均值方差原则，提出了模糊随机λ-均值方差模型，定义了λ-均值方差有效前沿和λ-均值方差有效解的概念，并讨论了位于不同的λ-均值方差有效前沿上的有效解之间的关系．由于不再要求投资者对未来有同质预期，因此，对不同的投资者建立了不同的资产组合选择模型，投资者根据他们对股票收益的不同预期产生自己的最优投资策略．此外，还收集了相关数据对提出的模型进行了实证分析．
(3)非标准类型的投资者往往会考虑除了投资收益和风险之外的其它目标，如流动性．本文中假设投资者同时考虑投资收益，风险和流动性三个目标，提出了模糊随机环境下的带复杂约束条件的多目标资产组合选择模型，并设计了基于妥协方法的遗传算法求解该多目标资产组合选择模型，从而产生投资者的一个妥协投资策略．
(4)针对已有的模糊随机机会约束多目标规划模型，本文给出了对一类特殊的模糊随机变量模糊随机机会约束多目标线性规划的确定等价模型，并利用交互式满意算法给出了求解该确定等价模型的传统求解算法．同时，为求解一般的模糊随机机会约束多目标规划模型，综合利用了模糊随机模拟和基于妥协方法的遗传算法提出了新的混合智能算法以获得决策者的一个妥协解．
(5)参照随机规划中的概率最大模型，利用模糊随机事件的本原机会测度的概念本文提出了模糊随机机会最大多目标规划模型．对一类特殊类型的模糊随机变量，本文给出了模糊随机机会最大多目标线性规划模型的确定等价模型，并给出了找到该确定等价模型的一个弱有效解的方法．此外，为求解一般的模糊随机机会最大多目标规划模型，本文提出了结合模糊随机模拟和妥协遗传算法的混合智能算法以产生一个妥协解．
(6)参照随机资产组合选择问题的两种安全第一模型的形式，提出了模糊随机机会约束资产组合选择模型和模糊随机机会最大资产组合选择模型．收集了相关数据对以上两个模型进行了实证分析．
(7)从区间序的角度考虑了区间目标规划模型，将该模型最终转化为一个线性规划问题进行求解．同时，以资产组合的期望收益率的绝对偏差函数度量投资风险，利用区间数表示各股票的期望收益率和绝对偏差，提出了一类基于区间目标规划的资产组合选择模型，并收集了相关数据进行了实证分析．
(8)将本文中提出的这些模糊随机环境下的资产组合选择模型与已有的随机资产组合选择模型和模糊资产组合选择模型进行了比较分析．
以模糊随机理论为核心，本文对模糊随机多目标规划模型，传统求解算法以及混合智能算法进行了分析和探讨．同时，紧密结合现有的关于模糊随机多目标规划的研究成果，对在模糊随机环境下的资产组合选择问题进行了模型，算法和实证分析上的讨论．本文的研究工作无疑对模糊随机多目标规划和在模糊随机环境下的资产组合选择问题的研究起到了积极的推动作用．
Among types of uncertainty surrounding real life problems, randomness (stochastic variation) and fuzziness (vagueness) play a pivotal role. Randomness is one type of uncertainty that describes occurrence of affairs, and random variables are used to describe the stochastic phenomena. Fuzzyness is one type of uncertainty that describes the uncertain state of affairs, and fuzzy sets are used to describe fuzzy phenomena. Accordingly, stochastic programming and fuzzy programming have been proposed to make decision under uncertainty environment. However, in a decision-making process, we may face a hybrid uncertain environment where fuzziness and randomness coexist. In such cases, the concept of fuzzy random variable is a useful tool dealing with the two types of uncertainty simultaneously. Roughly speaking, a fuzzy random variable is a measurable function from a probability space to the set of fuzzy variables. In other words, a fuzzy random variable is a random variable taking fuzzy values. Fuzzy random phenomena are existed extensively in real life problems. Therefore, research on fuzzy random multiobjective decision making bear not only academic but also practical significance.
Portfolio selection theory deals with how to distribute investment money among different assets to maximize return as well as minimize risk. There are many kinds of uncertainty including randomness and fuzziness simultaneously in the stock market. So far, on the assumption that the future return rate of each securities is a random variable, most of the portfolio selection models are based on probability theory. Recently, fuzziness existed in the stock market are gradually recognized by some scholars. Several fuzzy portfolio selection models have been proposed on the assumption that the return rate of each securities is a fuzzy variable. Actually, either the stochastic portfolio selection models or fuzzy portfolio selection models consider only one type of uncertainty. Therefore, portfolio selection under hybrid uncertain environment needs further research.
Hence, with summarizing the known researches on fuzzy random programming and portfolio selection models, we discuss the fuzzy random multiobjective programming models and algorithms based on the fuzzy random theory in this dissertation. By using fuzzy random variable to describe the fuzzy random phenomena existed in the stock market, we propose several portfolio selection models as well as multiobjective portfolio selection models under fuzzy random environment. The major achievements in this dissertation are listed as follows:
(1) In order to characterize the randomness and fuzziness existed in the stock market simultaneously, the return rate of each securities is treated as a fuzzy random variable. The historical data, the experts' knowledge and experience as well as the investor's individual expectation about the return rate of each securities are considered in this disseration.
(2) Following the mean variance principle, the fuzzy randomλ-mean variance model is proposed. Theλ-mean variance efficient frontier and theλ-mean variance efficient solution are defined, and the relations between theλ-mean variance efficient solutions located on differentλ-mean variance efficient frontiers are also discussed. The same expectation assumption on the future return rate of each securities, which is a basic assumption in Markowitz's mean-variance model, is no longer necessary, different investors can built their different portfolio selection models and therefore obtain their optimal investment strategy from the the proposed fuzzy randomλ-mean variance model. Furthermore, with the collection of historical data practical analysis is carried out for the proposed portfolio selection model.
(3) Non-standard investors always consider more objectives besides return and risk, such as liquidity. Suppose that a investor consider return, risk and liquidity simultaneously, a constrained multiobjective portfolio selection model under fuzzy random environment is proposed, and the compromised-based genetic algorithm is used to obtain a compromise investment strategy.
(4) Based on the fuzzy random chance constrained multiobjective programming model proposed by Liu, some crisp equivalent models are given for some special kinds of fuzzy random variables. For the general kinds of fuzzy random variables, hybrid intelligent algorithms which combine fuzzy random simulation and compromised-based genetic algorithms are combined together to propose a hybrid intelligent algorithm and therefore a compromised solution can be obtained.
(5) Similar to the probability maximization model in stochastic programming, a class of fuzzy random chance maximization multiobjective programming model based on the primitive chance of fuzzy random event is proposed. For a special kind of fuzzy random variables, the crisp equivalent models are proposed and a method is also proposed to obtain a weakly efficient solution. For general kinds of fuzzy random variables, hybrid intelligent algorithm which combines fuzzy random simulation and compromised-based genetic algorithms is also designed to obtain a compromised solution.
(6) Similar to the safety first model, fuzzy random chance maximization portfolio selection model and fuzzy random chance constrained portfolio selection model are proposed. Moreover, with the collection of historical data practical analysis is carried out to verify the effectiveness of the proposed models.
(7) The interval goal programming is discussed from the view of interval orders and it is converted into a linear programming problem at last. By using the absolute deviation risk function, a portfolio selection model with interval return and interval absolute deviation is proposed based on the interval goal programming model. Moreover, with the collection of historical data practical analysis is carried out to verify the effectiveness of the proposed model.
(8) The fuzzy random portfolio selection models proposed in this dissertation are compared with some of the existing stochastic and fuzzy portfolio selection models in literature.
Based on fuzzy random theory, the fuzzy random multiobjective programming models, its traditional algorithms and hybrid intelligent algorithms are discussed in this dissertation. Moreover, the fuzzy random portfolio selection models, its algorithms and practical analysis are also discussed. Undoubtedly, these researches included in the dissertation are helpful to develop, improve and prompt the researches on fuzzy random multiobjective programming and fuzzy random portfolio selection models.
资产组合选择理论研究如何把投资资金分配到不同的资产中，以达到分散风险并确保收益的目的．在证券市场中存在着多种不确定性，既有随机性也有模糊性．到目前为止，多数资产组合选择模型都是以股票的未来收益率是随机变量为前提并建立在概率论基础上的．最近，证券市场中存在的模糊不确定性逐渐被人们认识到，在假设股票的未来收益率为模糊变量的条件下，各种模糊资产组合选择模型被相继提出．模糊资产组合选择问题的研究成为一个新的极有前途的研究方向．实际上，无论是随机或模糊资产组合选择模型，它们往往只考虑了一种不确定性，对在双重不确定环境下的资产组合选择问题需要做进一步的研究．
为此，本文在广泛借鉴和吸收国内外研究成果的基础上，以模糊随机理论为基础，对模糊随机多目标规划模型和算法进行了研究．同时，利用模糊随机变量刻画证券市场中存在的模糊随机现象，建立了若干在模糊随机环境下的资产组合选择模型(其中包括模糊随机环境下的多目标资产组合选择模型)．主要工作如下：
(1)为刻画证券市场中同时出现的随机不确定性和模糊不确定性，本文将股票的未来收益率处理为模糊随机变量，同时考虑了历史交易数据，专家的经验和知识以及投资者个人对各股票未来收益的预期三方面因素．
(2)遵循Markowitz提出的均值方差原则，提出了模糊随机λ-均值方差模型，定义了λ-均值方差有效前沿和λ-均值方差有效解的概念，并讨论了位于不同的λ-均值方差有效前沿上的有效解之间的关系．由于不再要求投资者对未来有同质预期，因此，对不同的投资者建立了不同的资产组合选择模型，投资者根据他们对股票收益的不同预期产生自己的最优投资策略．此外，还收集了相关数据对提出的模型进行了实证分析．
(3)非标准类型的投资者往往会考虑除了投资收益和风险之外的其它目标，如流动性．本文中假设投资者同时考虑投资收益，风险和流动性三个目标，提出了模糊随机环境下的带复杂约束条件的多目标资产组合选择模型，并设计了基于妥协方法的遗传算法求解该多目标资产组合选择模型，从而产生投资者的一个妥协投资策略．
(4)针对已有的模糊随机机会约束多目标规划模型，本文给出了对一类特殊的模糊随机变量模糊随机机会约束多目标线性规划的确定等价模型，并利用交互式满意算法给出了求解该确定等价模型的传统求解算法．同时，为求解一般的模糊随机机会约束多目标规划模型，综合利用了模糊随机模拟和基于妥协方法的遗传算法提出了新的混合智能算法以获得决策者的一个妥协解．
(5)参照随机规划中的概率最大模型，利用模糊随机事件的本原机会测度的概念本文提出了模糊随机机会最大多目标规划模型．对一类特殊类型的模糊随机变量，本文给出了模糊随机机会最大多目标线性规划模型的确定等价模型，并给出了找到该确定等价模型的一个弱有效解的方法．此外，为求解一般的模糊随机机会最大多目标规划模型，本文提出了结合模糊随机模拟和妥协遗传算法的混合智能算法以产生一个妥协解．
(6)参照随机资产组合选择问题的两种安全第一模型的形式，提出了模糊随机机会约束资产组合选择模型和模糊随机机会最大资产组合选择模型．收集了相关数据对以上两个模型进行了实证分析．
(7)从区间序的角度考虑了区间目标规划模型，将该模型最终转化为一个线性规划问题进行求解．同时，以资产组合的期望收益率的绝对偏差函数度量投资风险，利用区间数表示各股票的期望收益率和绝对偏差，提出了一类基于区间目标规划的资产组合选择模型，并收集了相关数据进行了实证分析．
(8)将本文中提出的这些模糊随机环境下的资产组合选择模型与已有的随机资产组合选择模型和模糊资产组合选择模型进行了比较分析．
以模糊随机理论为核心，本文对模糊随机多目标规划模型，传统求解算法以及混合智能算法进行了分析和探讨．同时，紧密结合现有的关于模糊随机多目标规划的研究成果，对在模糊随机环境下的资产组合选择问题进行了模型，算法和实证分析上的讨论．本文的研究工作无疑对模糊随机多目标规划和在模糊随机环境下的资产组合选择问题的研究起到了积极的推动作用．
Among types of uncertainty surrounding real life problems, randomness (stochastic variation) and fuzziness (vagueness) play a pivotal role. Randomness is one type of uncertainty that describes occurrence of affairs, and random variables are used to describe the stochastic phenomena. Fuzzyness is one type of uncertainty that describes the uncertain state of affairs, and fuzzy sets are used to describe fuzzy phenomena. Accordingly, stochastic programming and fuzzy programming have been proposed to make decision under uncertainty environment. However, in a decision-making process, we may face a hybrid uncertain environment where fuzziness and randomness coexist. In such cases, the concept of fuzzy random variable is a useful tool dealing with the two types of uncertainty simultaneously. Roughly speaking, a fuzzy random variable is a measurable function from a probability space to the set of fuzzy variables. In other words, a fuzzy random variable is a random variable taking fuzzy values. Fuzzy random phenomena are existed extensively in real life problems. Therefore, research on fuzzy random multiobjective decision making bear not only academic but also practical significance.
Portfolio selection theory deals with how to distribute investment money among different assets to maximize return as well as minimize risk. There are many kinds of uncertainty including randomness and fuzziness simultaneously in the stock market. So far, on the assumption that the future return rate of each securities is a random variable, most of the portfolio selection models are based on probability theory. Recently, fuzziness existed in the stock market are gradually recognized by some scholars. Several fuzzy portfolio selection models have been proposed on the assumption that the return rate of each securities is a fuzzy variable. Actually, either the stochastic portfolio selection models or fuzzy portfolio selection models consider only one type of uncertainty. Therefore, portfolio selection under hybrid uncertain environment needs further research.
Hence, with summarizing the known researches on fuzzy random programming and portfolio selection models, we discuss the fuzzy random multiobjective programming models and algorithms based on the fuzzy random theory in this dissertation. By using fuzzy random variable to describe the fuzzy random phenomena existed in the stock market, we propose several portfolio selection models as well as multiobjective portfolio selection models under fuzzy random environment. The major achievements in this dissertation are listed as follows:
(1) In order to characterize the randomness and fuzziness existed in the stock market simultaneously, the return rate of each securities is treated as a fuzzy random variable. The historical data, the experts' knowledge and experience as well as the investor's individual expectation about the return rate of each securities are considered in this disseration.
(2) Following the mean variance principle, the fuzzy randomλ-mean variance model is proposed. Theλ-mean variance efficient frontier and theλ-mean variance efficient solution are defined, and the relations between theλ-mean variance efficient solutions located on differentλ-mean variance efficient frontiers are also discussed. The same expectation assumption on the future return rate of each securities, which is a basic assumption in Markowitz's mean-variance model, is no longer necessary, different investors can built their different portfolio selection models and therefore obtain their optimal investment strategy from the the proposed fuzzy randomλ-mean variance model. Furthermore, with the collection of historical data practical analysis is carried out for the proposed portfolio selection model.
(3) Non-standard investors always consider more objectives besides return and risk, such as liquidity. Suppose that a investor consider return, risk and liquidity simultaneously, a constrained multiobjective portfolio selection model under fuzzy random environment is proposed, and the compromised-based genetic algorithm is used to obtain a compromise investment strategy.
(4) Based on the fuzzy random chance constrained multiobjective programming model proposed by Liu, some crisp equivalent models are given for some special kinds of fuzzy random variables. For the general kinds of fuzzy random variables, hybrid intelligent algorithms which combine fuzzy random simulation and compromised-based genetic algorithms are combined together to propose a hybrid intelligent algorithm and therefore a compromised solution can be obtained.
(5) Similar to the probability maximization model in stochastic programming, a class of fuzzy random chance maximization multiobjective programming model based on the primitive chance of fuzzy random event is proposed. For a special kind of fuzzy random variables, the crisp equivalent models are proposed and a method is also proposed to obtain a weakly efficient solution. For general kinds of fuzzy random variables, hybrid intelligent algorithm which combines fuzzy random simulation and compromised-based genetic algorithms is also designed to obtain a compromised solution.
(6) Similar to the safety first model, fuzzy random chance maximization portfolio selection model and fuzzy random chance constrained portfolio selection model are proposed. Moreover, with the collection of historical data practical analysis is carried out to verify the effectiveness of the proposed models.
(7) The interval goal programming is discussed from the view of interval orders and it is converted into a linear programming problem at last. By using the absolute deviation risk function, a portfolio selection model with interval return and interval absolute deviation is proposed based on the interval goal programming model. Moreover, with the collection of historical data practical analysis is carried out to verify the effectiveness of the proposed model.
(8) The fuzzy random portfolio selection models proposed in this dissertation are compared with some of the existing stochastic and fuzzy portfolio selection models in literature.
Based on fuzzy random theory, the fuzzy random multiobjective programming models, its traditional algorithms and hybrid intelligent algorithms are discussed in this dissertation. Moreover, the fuzzy random portfolio selection models, its algorithms and practical analysis are also discussed. Undoubtedly, these researches included in the dissertation are helpful to develop, improve and prompt the researches on fuzzy random multiobjective programming and fuzzy random portfolio selection models.
引文
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