几类特殊模糊集的理论与应用研究
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摘要
通过引入隶属度函数μA(x),非隶属度函数vA(x)和犹豫度函数πA(x),Atanassov建立了直觉模糊集,它推广了Zadeh模糊集理论。根据直觉模糊集定义,μA(x)表示支持派的比例,vA(x)表示反对派的比例,πA(x)表示弃权派的比例。虽然直觉模糊集诞生后,众多学者开始研究该领域,并把它广泛地应用到决策分析、模式识别、医疗诊断、自动控制和模糊推理等研究领域。然而,传统的研究并未涉及对弃权派可能发展趋势的分析,因此,现有方法适合于静态模型而不适合动态模型。于是,徐泽水和Yager于2008年提出了一种动态决策模型,此后卫贵武和Su等也研究了动态决策模型。但是即便如此,他们也未对弃权派进行分析。而且考虑到在实践决策中,弃权派可能随着时间的迁移发生变化,因此在未来的动态决策研究中,弃权派将可能扮演重要的角色。有鉴于此,我们通过对犹豫度的分析,提出了一系列带参数的动态模糊集模型。
首先,我们假设对于事件A,μA(x)是坚定的支持派,vA(x)是坚定的反对派,πA(x)是最大的弃权派,πA*(x)=(1-λA0(x))πA(x))πA(x)是坚定的弃权派,πA(x)-πA*(x)=λA0(x)πA(x)是可转化的弃权派。在此,λAO(x)表示弃权派中可转化的比率。很显然三派总和μA(x)+vA(x)+πA(x)=1且0≤λAO(x)≤1。接着,我们又把可转化的弃权派进行划分,其中λA0(x)λA1(x)πA(x)转变成支持派,λA0(x)(1-λA1(x))πA(x)转变成反对派,在此,可转化的弃权派转变成支持派的比例为λAI(x),转变成反对派的比例为1-λAI(x),我们也有0≤λAI(x)≤1。根据对弃权派的分解,我们提出了一系列定义：带参数模糊集,带参数直觉模糊集,带参数区间值直觉模糊集等,并提出其构造方法。
其次,我们提出了一种新的广义区间值直觉模糊集模型,并证明了模糊集、区间值模糊集、直觉模糊集、区间值直觉模糊集、Vague集和区间值Vague集都是其特例。接着,我们证明对于建立在新广义区间值直觉模糊集之上的交集运算、并集运算和补集运算构成封闭的软代数系统,这个性质与模糊集的性质是一样的。
第三,基于直觉模糊集,我们提出了一些新的比率距离测度和计分函数,并指出这些测度与函数具备良好的数学性质。相应地,我们把这些距离测度和计分函数推广到带参数模糊集领域,并进而将其运用到模式识别与医疗诊断。仿真实验表明,相应的带参数模糊集方法比传统的模糊集方法更有效。
第四,我们提出了一些新的直觉模糊集和区间值直觉模糊集排序函数,并将其推广到带参数模糊集、带参数直觉模糊集、带参数区间值直觉模糊集和广义区间值直觉模糊集。我们证明了任意直觉模糊集的排序函数都是其带参数模型下的特例。最后,利用上述排序函数,我们将带参数模糊集、带参数直觉模糊集、带参数区间值直觉模糊集都运用到多属性决策。试验结果表明,我们可以通过调节参数到合适的数值而得到所需要的结果。因此,带参数模糊集方法能够被运用到动态决策领域,我们可以通过调节参数,从而调节隶属度函数、非隶属度函数和犹豫度函数,并进而预测所有可能的决策结果。
总之,本文提出的新方法扩展了模糊集和区间值模糊集在模式识别与决策中的应用。仿真实验结果表明,本文所提出的新方法比传统的模糊集方法和直觉模糊集方法更广泛、也更灵活。
By introducing membership function μA(x), non-membership function vA(x) and hesitancy function πA(x), the intuitionistic fuzzy sets (IFS) theory is established by Atanassov, which generalizes Zadeh's fuzzy sets (FS). According to the IFS definition, μA(x) denotes the proportion of the support party, vA(x) denotes the proportion of the opposition party, and πA(x) denotes the proportion of the absent party. Though many scholars studied IFS and applied it widely to decision making analysis, pattern recognition, medical diagnosis, automatic control, fuzzy reasoning, etc. However, traditional researches based on IFS do not consider the detachment of the absent party, which means that the absent party is not specifically analyzed in conventional model of IFS. Therefore, these methods are suitable for static model and unsuitable for dynamic model. Then Xu and Yager presented a dynamic decision making model in2008, which was also studied by Wei, Su, et al. But they also do not study the detachment of the absent party. Moreover, the absent party may change over time in practice, while conventional IFS method cannot deal with this kind of dynamic model. Thus, the research on the variation of absent party will play an important role in dynamic decision making, dynamic pattern recognition, dynamic automatic control and dynamic fuzzy reasoning. Taking into account this, we present a series of fuzzy sets models with parameters by analyzing the hesitancy function.
First, we assume that μA(x) is the firm support party of event A, vA(x) is the firm opposition party of event A, πA(x) is the maximum absent party of event A, πA*(x)=(1-λA0(x)) πA(x) is the firm absent party of event A, and πA(x)-πA*(x)=λA0(x)πA(x) denotes the convertible absent part, where λA0(x) is the proportion of the convertible absent individuals in all the absent individuals. Obviously we have μA(x)+vA(x)+πA(x)=1and0≤λA0(x)≤1. We divide the convertible absent part into two parts:λA0(x)λA1(x)πA(x) being the absent party which can be converted into the support party, and λA0(x)(1-λA1(x))πA(x) being the absent party which can be converted into the opposition party, where λA1(x) is the proportion of the convertible absent individuals being converted to the support party, and1-XλA1(x) is the proportion of the convertible absent individuals being converted to the opposition party. And we also have0≤λA1(x)≤l. According to the detachment of the absent party, we introduce a series of definitions:such as fuzzy sets with parameters (FSP), intuitionistic fuzzy sets with parameters (IFSP), interval-valued intuitionistic fuzzy sets with parameters (IVIFSP), etc. And then we also propose the construction methods of them.
Secondly, we present a novel generalized interval-valued intuitionistic fuzzy sets model (GIVIFS), which is proved to be the generalization of fuzzy sets (FS), interval-valued fuzzy sets (IVFS), intuitionistic fuzzy sets (IFS), interval-valued intuitionistic fuzzy sets (IVIFS), vague sets (VS), interval-valued vague sets (IVVS). Moreover, the GIVIFS model is proved to be a closed soft algebra system for the intersection operator, the join operator and the complement operator as fuzzy sets model.
Thirdly, we present some ratio distance measures and some novel score functions on IFS. And we study the properties of these measures and functions. Accordingly, we introduce the corresponding distance measures and score functions on FSP, IFSP, and IVIFSP. And then we apply them to pattern recognition and medical diagnosis, the simulation results show that the corresponding method with parameters is more effective than the conventional fuzzy sets method.
Fourthly, we present some novel ranking function on IFS and IVIFS, and generalized them to FSP, IFSP, IVIFSP and GIVIFSP. And we prove that any ranking function of IFS is the special case of the ranking function of IFSDP. Finally, taking advantage of the ranking functions above, FSP model, IFSP model and IVIFSP model are applied to multiple attribute decision making. The experimental results show that we can adjust the parameters to appropriate values to obtain all feasible results. Therefore, the FS method with parameters can be applied to the dynamic decision making field, and we can predict all the possible decision making results in the future according to the variation of membership function, non-membership function, and hesitancy function.
All in all, the new method proposed in this paper can expand the scope of FS and IVFS applied to pattern recognition and decision making. The simulation results show that the methods introduced in this paper are more comprehensive and flexible than the conventional FS method and IFS method.
首先,我们假设对于事件A,μA(x)是坚定的支持派,vA(x)是坚定的反对派,πA(x)是最大的弃权派,πA*(x)=(1-λA0(x))πA(x))πA(x)是坚定的弃权派,πA(x)-πA*(x)=λA0(x)πA(x)是可转化的弃权派。在此,λAO(x)表示弃权派中可转化的比率。很显然三派总和μA(x)+vA(x)+πA(x)=1且0≤λAO(x)≤1。接着,我们又把可转化的弃权派进行划分,其中λA0(x)λA1(x)πA(x)转变成支持派,λA0(x)(1-λA1(x))πA(x)转变成反对派,在此,可转化的弃权派转变成支持派的比例为λAI(x),转变成反对派的比例为1-λAI(x),我们也有0≤λAI(x)≤1。根据对弃权派的分解,我们提出了一系列定义：带参数模糊集,带参数直觉模糊集,带参数区间值直觉模糊集等,并提出其构造方法。
其次,我们提出了一种新的广义区间值直觉模糊集模型,并证明了模糊集、区间值模糊集、直觉模糊集、区间值直觉模糊集、Vague集和区间值Vague集都是其特例。接着,我们证明对于建立在新广义区间值直觉模糊集之上的交集运算、并集运算和补集运算构成封闭的软代数系统,这个性质与模糊集的性质是一样的。
第三,基于直觉模糊集,我们提出了一些新的比率距离测度和计分函数,并指出这些测度与函数具备良好的数学性质。相应地,我们把这些距离测度和计分函数推广到带参数模糊集领域,并进而将其运用到模式识别与医疗诊断。仿真实验表明,相应的带参数模糊集方法比传统的模糊集方法更有效。
第四,我们提出了一些新的直觉模糊集和区间值直觉模糊集排序函数,并将其推广到带参数模糊集、带参数直觉模糊集、带参数区间值直觉模糊集和广义区间值直觉模糊集。我们证明了任意直觉模糊集的排序函数都是其带参数模型下的特例。最后,利用上述排序函数,我们将带参数模糊集、带参数直觉模糊集、带参数区间值直觉模糊集都运用到多属性决策。试验结果表明,我们可以通过调节参数到合适的数值而得到所需要的结果。因此,带参数模糊集方法能够被运用到动态决策领域,我们可以通过调节参数,从而调节隶属度函数、非隶属度函数和犹豫度函数,并进而预测所有可能的决策结果。
总之,本文提出的新方法扩展了模糊集和区间值模糊集在模式识别与决策中的应用。仿真实验结果表明,本文所提出的新方法比传统的模糊集方法和直觉模糊集方法更广泛、也更灵活。
By introducing membership function μA(x), non-membership function vA(x) and hesitancy function πA(x), the intuitionistic fuzzy sets (IFS) theory is established by Atanassov, which generalizes Zadeh's fuzzy sets (FS). According to the IFS definition, μA(x) denotes the proportion of the support party, vA(x) denotes the proportion of the opposition party, and πA(x) denotes the proportion of the absent party. Though many scholars studied IFS and applied it widely to decision making analysis, pattern recognition, medical diagnosis, automatic control, fuzzy reasoning, etc. However, traditional researches based on IFS do not consider the detachment of the absent party, which means that the absent party is not specifically analyzed in conventional model of IFS. Therefore, these methods are suitable for static model and unsuitable for dynamic model. Then Xu and Yager presented a dynamic decision making model in2008, which was also studied by Wei, Su, et al. But they also do not study the detachment of the absent party. Moreover, the absent party may change over time in practice, while conventional IFS method cannot deal with this kind of dynamic model. Thus, the research on the variation of absent party will play an important role in dynamic decision making, dynamic pattern recognition, dynamic automatic control and dynamic fuzzy reasoning. Taking into account this, we present a series of fuzzy sets models with parameters by analyzing the hesitancy function.
First, we assume that μA(x) is the firm support party of event A, vA(x) is the firm opposition party of event A, πA(x) is the maximum absent party of event A, πA*(x)=(1-λA0(x)) πA(x) is the firm absent party of event A, and πA(x)-πA*(x)=λA0(x)πA(x) denotes the convertible absent part, where λA0(x) is the proportion of the convertible absent individuals in all the absent individuals. Obviously we have μA(x)+vA(x)+πA(x)=1and0≤λA0(x)≤1. We divide the convertible absent part into two parts:λA0(x)λA1(x)πA(x) being the absent party which can be converted into the support party, and λA0(x)(1-λA1(x))πA(x) being the absent party which can be converted into the opposition party, where λA1(x) is the proportion of the convertible absent individuals being converted to the support party, and1-XλA1(x) is the proportion of the convertible absent individuals being converted to the opposition party. And we also have0≤λA1(x)≤l. According to the detachment of the absent party, we introduce a series of definitions:such as fuzzy sets with parameters (FSP), intuitionistic fuzzy sets with parameters (IFSP), interval-valued intuitionistic fuzzy sets with parameters (IVIFSP), etc. And then we also propose the construction methods of them.
Secondly, we present a novel generalized interval-valued intuitionistic fuzzy sets model (GIVIFS), which is proved to be the generalization of fuzzy sets (FS), interval-valued fuzzy sets (IVFS), intuitionistic fuzzy sets (IFS), interval-valued intuitionistic fuzzy sets (IVIFS), vague sets (VS), interval-valued vague sets (IVVS). Moreover, the GIVIFS model is proved to be a closed soft algebra system for the intersection operator, the join operator and the complement operator as fuzzy sets model.
Thirdly, we present some ratio distance measures and some novel score functions on IFS. And we study the properties of these measures and functions. Accordingly, we introduce the corresponding distance measures and score functions on FSP, IFSP, and IVIFSP. And then we apply them to pattern recognition and medical diagnosis, the simulation results show that the corresponding method with parameters is more effective than the conventional fuzzy sets method.
Fourthly, we present some novel ranking function on IFS and IVIFS, and generalized them to FSP, IFSP, IVIFSP and GIVIFSP. And we prove that any ranking function of IFS is the special case of the ranking function of IFSDP. Finally, taking advantage of the ranking functions above, FSP model, IFSP model and IVIFSP model are applied to multiple attribute decision making. The experimental results show that we can adjust the parameters to appropriate values to obtain all feasible results. Therefore, the FS method with parameters can be applied to the dynamic decision making field, and we can predict all the possible decision making results in the future according to the variation of membership function, non-membership function, and hesitancy function.
All in all, the new method proposed in this paper can expand the scope of FS and IVFS applied to pattern recognition and decision making. The simulation results show that the methods introduced in this paper are more comprehensive and flexible than the conventional FS method and IFS method.
引文
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