区间值模糊集之间几种距离的研究
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摘要
本文主要讨论了区间值模糊集之间的几种距离并研究了相应的区间值模糊数距离空间的性质。主要工作如下：
(1)给出了定义在实数域R上的区间值模糊集之间的三种距离d~*，d_p~*和d_∞~*，讨论了它们的基本性质，如齐次性、平移不变性等。在这三种不同距离意义下分别讨论了相应的区间值模糊数的距离空间(IF~*(R)，d~*)，(IF~*(R)，d_p~*)和(IF~*(R)，d_∞~*)的完备性和可分性。我们证明了当K是R上的非空紧子集时，紧支撑包含在K中的所有区间值模糊数所成之集TG~*(K)((?)IF*(R))分别关于d~*和d_p~*是完备子空间。对于距离d_∞~*全空间(IF~*(R)，d_∞~*)是完备的。同时我们分别举例说明全空间(IF~*(R)，d~*)和(IF~*(R)，d_p~*)不是完备的，(IF~*(R)，d~*)和(IF~*(R)，d_∞~*)不是可分的。我们还对区间值模糊数序列在这三种距离意义下的收敛性之间的关系进行了讨论，得到一些结果。
(2)我们对基于Hausdorff度量意义下的两种离散型区间值模糊集的距离进行了推广，同时给出了一种新的基于Hausdorff度量意义下的离散型区间值模糊集之间的距离，并研究了它们的性质；对这三种离散型距离进行了比较分析，得出了它们之间的大小关系。
In this dissertation some researches on distancs of interval-valued fuzzy sets and metric space with respect to the distance are made. The main results as follow:
(1) We give three kinds of distance of interval-valued fuzzy sets defined on real line R, denoted by d~*,d_p~* and d_∞~*, and study some properties, for example, homogeneity, translation invariant and so on. We investigate the completeness and separability of interval-valued fuzzy numbers metric spaces (IF~*(R),d~*), (IF~*(R),d_p~*) and (IF~*(R),d_∞~*). We show that the whole space (IF~*(R), d_∞~*) is complete metric space, and when K is a nonempty compact subset of R, the metric space (IF~*(K),d~*) and (IF~*(K),d_p~*) are complete. We discuss the relation between the convergence of sequence of interval-valued fuzzy numbers in the sence of the distance d~*, d_p~* and d_∞~*.
(2) We generalize the concepts of two kinds of discrete distance of interval-valued fuzzy sets based on Hausdorff metric, and introduce a new distance of discrete interval-valued fuzzy sets based on Hausdorff metric and study some properties. We compare these three kinds of distance and give the max-min relation among them.
(1)给出了定义在实数域R上的区间值模糊集之间的三种距离d~*，d_p~*和d_∞~*，讨论了它们的基本性质，如齐次性、平移不变性等。在这三种不同距离意义下分别讨论了相应的区间值模糊数的距离空间(IF~*(R)，d~*)，(IF~*(R)，d_p~*)和(IF~*(R)，d_∞~*)的完备性和可分性。我们证明了当K是R上的非空紧子集时，紧支撑包含在K中的所有区间值模糊数所成之集TG~*(K)((?)IF*(R))分别关于d~*和d_p~*是完备子空间。对于距离d_∞~*全空间(IF~*(R)，d_∞~*)是完备的。同时我们分别举例说明全空间(IF~*(R)，d~*)和(IF~*(R)，d_p~*)不是完备的，(IF~*(R)，d~*)和(IF~*(R)，d_∞~*)不是可分的。我们还对区间值模糊数序列在这三种距离意义下的收敛性之间的关系进行了讨论，得到一些结果。
(2)我们对基于Hausdorff度量意义下的两种离散型区间值模糊集的距离进行了推广，同时给出了一种新的基于Hausdorff度量意义下的离散型区间值模糊集之间的距离，并研究了它们的性质；对这三种离散型距离进行了比较分析，得出了它们之间的大小关系。
In this dissertation some researches on distancs of interval-valued fuzzy sets and metric space with respect to the distance are made. The main results as follow:
(1) We give three kinds of distance of interval-valued fuzzy sets defined on real line R, denoted by d~*,d_p~* and d_∞~*, and study some properties, for example, homogeneity, translation invariant and so on. We investigate the completeness and separability of interval-valued fuzzy numbers metric spaces (IF~*(R),d~*), (IF~*(R),d_p~*) and (IF~*(R),d_∞~*). We show that the whole space (IF~*(R), d_∞~*) is complete metric space, and when K is a nonempty compact subset of R, the metric space (IF~*(K),d~*) and (IF~*(K),d_p~*) are complete. We discuss the relation between the convergence of sequence of interval-valued fuzzy numbers in the sence of the distance d~*, d_p~* and d_∞~*.
(2) We generalize the concepts of two kinds of discrete distance of interval-valued fuzzy sets based on Hausdorff metric, and introduce a new distance of discrete interval-valued fuzzy sets based on Hausdorff metric and study some properties. We compare these three kinds of distance and give the max-min relation among them.
引文
[1] L.A.Zadeh, Fuzzy sets[J], Inforrnation and Control 8(1965)338-353.
[2] 吴丛忻、马明，模糊分析学基础[M]，国防工业出版社，1991．
[3] 杨纶标、高英仪，模糊数学原理及应用[M]，华南理工大学出版社，2003．
[4] Wang.G and Li.X, The applications of interval-valued fuzzy numbers and interval-distribution numbers[J], Fuzzy Sets and Systems 98(1998)331-335.
[5] 胡宝清，模糊理论基础[M]，武汉大学出版社，2004．
[6] Wang.G and Li.X, Correlation and information energy of interval-valued fuzzy numbers[J], Fuzzy Sets and Systems 103(1999)169-175.
[7] 程其襄等，实变函数与泛函分析基础[M]，高等教育出版社，1983．
[8] G. Meng, Interval Valued L-fuzzy set and its decomposition theorem[J], BUSEFAL 41(1989)23-30.
[9] P.Grzegorzewski, Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric[J], Fuzzy Sets and Systems 148(2004)319-328.
[10] 李雷、吴丛忻，集值分析[M]，科学出版社，2003．
[11] 张文修、王国俊、刘旺金等，模糊数学引论[M]，西安交通大学出版社，1991．
[12] P. Diamond and P.E.Kloeden, Metric spaces of fuzzy sets, Theory and Applications [M], World Scientific, Singapore, 1994.
[13] B. Gorzafczany, Approximate inference with interval-valued fuzzy sets-an outline[J], in: proc Polish Symp on Interval and Fuzzy Math Poznan Poland (1983)89-95.
[14] G.Meng, The basic theories of interval-valued fuzzy sets[J], Mathrnatice Applicate 6(2)(1993)212-217.
[15] B.Turksen, Interval valud fuzzy sets based on norrnal forrns[J], Fuzzy Sets and Systems 2o(1986)191-210.
[16] E.Szrnidt and J.Kacprzyk, Distances between intuitionistic fuzzy sets[J], Fuzzy Sets and Systems 114(2000)505-518.
[17] E.Szrnidt and J.Kacprzyk, On rneasuring distances between intuitionistic fuzzy sets[J], Note IFS 3(1997)1-13.
[18] K.Atanassov, Intuitionistic fuzzy sets[J], Fuzzy Sets and Systems 20(1986)87-96
[19] K.Atanassov, More on intuitionistic fuzzy sets[J], Fuzzy Sets and Systems 33(1989)37-46.
[20] K.Atanassov, Operators over interval valued intuitionistic fuzzy sets[J], Fuzzy Sets and Systems 64(1994)159-174.
[21] K.Atanassov, New operations defined over the intuitionistic fuzzy sets[J], Fuzzy Sets and Systems 61(1994)137-142.
[22] Dug.H, A note on the correlation and information energy of interval-valued fuzzy numbers[J], Fuzzy Sets and Systems 123(2001)89-92.
[23] L.Tran and L.Duckstein, Comparison of fuzzy numbers using a fuzzy distance measure[J], Fuzzy Sets and Systems 130(2002)331-341.
[24] P.Burillo and H.Bustince, Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets[J], Fuzzy Sets and Systems 78(1996)305-316.
[25] Wen-Liang Hung and Miin-Shen Yang, Similiary measure of intuitionistic fuzzy sets based on Hausdorff distance[J], Pattern Recognition Letters 25(2003) 1603-1611.
[26] Zhizhen Liang and Pengfei Shi, Similiary measure on intuitionistic fuzzy sets[J], Pattern Recognition Letters 24(2003)2687-2693.
[27] Deng-Feng Li, Some measure of dissimilarity in intuitionistic fuzzy structures[J], Journal of Computer and System Sciences 68(2004)115-122.
[28] Li Dengfeng and Cheng Chuntian, New similiary measure of intuitionistic fuzzy sets and application to pattern recognitions[J], Pattern Recognition Letters 23(2002)221-225.
[29] 刘慧林、冯汝鹏，一种新的模糊数的距离定义[J]，模糊系统与数学17(2003)49—53．
[30] 刘慧林、冯汝鹏，新的模糊数的模糊距离[J]，模糊系统与数学19(2005)106-109．
[31] 颜云志、冯玉瑚，非紧模糊数的广义距离空间[J]，华东大学学报30(2004)21-25．
[32] 刘华文，基于距离测度的模糊数排序[J]，山东大学学报39(2004)30-39．
[33] 王莉、张立国，模糊数空间上下确界的度量刻划[J]，西北师范大学学报38(2002)26-27．
[34] J. Li, On Egoroff's theorems on fuzzy measure space[J], Fuzzy Sets and Systems 135(2003) 367-375.
[35] J. Li, Order continuity of monotone set function and convergence of measurable functions sequence[J], Applied Mathematics and Computation 135(2003) 211-218.
[36] J. Li, M. Yasuda, Egoroff's theorems on monotone non-additive measure space[J], Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems 12(2004) 61-68.
[37] J. Li, M. Yasuda, Lusin's theorem on fuzzy measure spaces[J], Fuzzy Sets and Systems 146(2004) 121-133.
[38] J. Song, J. Li, Regularity of null-additive fuzzy measure on metric spaces[J], International Journal General Systems 2003, Vol.33(3): 271-279.
[39] J. Song, J. Li, Lebesgue's theorerns in non-additive rneasure theory[J], Fuzzy Sets and Systems 149(2005) 543-548.
[2] 吴丛忻、马明，模糊分析学基础[M]，国防工业出版社，1991．
[3] 杨纶标、高英仪，模糊数学原理及应用[M]，华南理工大学出版社，2003．
[4] Wang.G and Li.X, The applications of interval-valued fuzzy numbers and interval-distribution numbers[J], Fuzzy Sets and Systems 98(1998)331-335.
[5] 胡宝清，模糊理论基础[M]，武汉大学出版社，2004．
[6] Wang.G and Li.X, Correlation and information energy of interval-valued fuzzy numbers[J], Fuzzy Sets and Systems 103(1999)169-175.
[7] 程其襄等，实变函数与泛函分析基础[M]，高等教育出版社，1983．
[8] G. Meng, Interval Valued L-fuzzy set and its decomposition theorem[J], BUSEFAL 41(1989)23-30.
[9] P.Grzegorzewski, Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric[J], Fuzzy Sets and Systems 148(2004)319-328.
[10] 李雷、吴丛忻，集值分析[M]，科学出版社，2003．
[11] 张文修、王国俊、刘旺金等，模糊数学引论[M]，西安交通大学出版社，1991．
[12] P. Diamond and P.E.Kloeden, Metric spaces of fuzzy sets, Theory and Applications [M], World Scientific, Singapore, 1994.
[13] B. Gorzafczany, Approximate inference with interval-valued fuzzy sets-an outline[J], in: proc Polish Symp on Interval and Fuzzy Math Poznan Poland (1983)89-95.
[14] G.Meng, The basic theories of interval-valued fuzzy sets[J], Mathrnatice Applicate 6(2)(1993)212-217.
[15] B.Turksen, Interval valud fuzzy sets based on norrnal forrns[J], Fuzzy Sets and Systems 2o(1986)191-210.
[16] E.Szrnidt and J.Kacprzyk, Distances between intuitionistic fuzzy sets[J], Fuzzy Sets and Systems 114(2000)505-518.
[17] E.Szrnidt and J.Kacprzyk, On rneasuring distances between intuitionistic fuzzy sets[J], Note IFS 3(1997)1-13.
[18] K.Atanassov, Intuitionistic fuzzy sets[J], Fuzzy Sets and Systems 20(1986)87-96
[19] K.Atanassov, More on intuitionistic fuzzy sets[J], Fuzzy Sets and Systems 33(1989)37-46.
[20] K.Atanassov, Operators over interval valued intuitionistic fuzzy sets[J], Fuzzy Sets and Systems 64(1994)159-174.
[21] K.Atanassov, New operations defined over the intuitionistic fuzzy sets[J], Fuzzy Sets and Systems 61(1994)137-142.
[22] Dug.H, A note on the correlation and information energy of interval-valued fuzzy numbers[J], Fuzzy Sets and Systems 123(2001)89-92.
[23] L.Tran and L.Duckstein, Comparison of fuzzy numbers using a fuzzy distance measure[J], Fuzzy Sets and Systems 130(2002)331-341.
[24] P.Burillo and H.Bustince, Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets[J], Fuzzy Sets and Systems 78(1996)305-316.
[25] Wen-Liang Hung and Miin-Shen Yang, Similiary measure of intuitionistic fuzzy sets based on Hausdorff distance[J], Pattern Recognition Letters 25(2003) 1603-1611.
[26] Zhizhen Liang and Pengfei Shi, Similiary measure on intuitionistic fuzzy sets[J], Pattern Recognition Letters 24(2003)2687-2693.
[27] Deng-Feng Li, Some measure of dissimilarity in intuitionistic fuzzy structures[J], Journal of Computer and System Sciences 68(2004)115-122.
[28] Li Dengfeng and Cheng Chuntian, New similiary measure of intuitionistic fuzzy sets and application to pattern recognitions[J], Pattern Recognition Letters 23(2002)221-225.
[29] 刘慧林、冯汝鹏，一种新的模糊数的距离定义[J]，模糊系统与数学17(2003)49—53．
[30] 刘慧林、冯汝鹏，新的模糊数的模糊距离[J]，模糊系统与数学19(2005)106-109．
[31] 颜云志、冯玉瑚，非紧模糊数的广义距离空间[J]，华东大学学报30(2004)21-25．
[32] 刘华文，基于距离测度的模糊数排序[J]，山东大学学报39(2004)30-39．
[33] 王莉、张立国，模糊数空间上下确界的度量刻划[J]，西北师范大学学报38(2002)26-27．
[34] J. Li, On Egoroff's theorems on fuzzy measure space[J], Fuzzy Sets and Systems 135(2003) 367-375.
[35] J. Li, Order continuity of monotone set function and convergence of measurable functions sequence[J], Applied Mathematics and Computation 135(2003) 211-218.
[36] J. Li, M. Yasuda, Egoroff's theorems on monotone non-additive measure space[J], Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems 12(2004) 61-68.
[37] J. Li, M. Yasuda, Lusin's theorem on fuzzy measure spaces[J], Fuzzy Sets and Systems 146(2004) 121-133.
[38] J. Song, J. Li, Regularity of null-additive fuzzy measure on metric spaces[J], International Journal General Systems 2003, Vol.33(3): 271-279.
[39] J. Song, J. Li, Lebesgue's theorerns in non-additive rneasure theory[J], Fuzzy Sets and Systems 149(2005) 543-548.