2012
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Cover vol. 9, no. 3, october 2012
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NEW MODELS AND ALGORITHMS FOR SOLUTIONS OF
SINGLESIGNED FULLY FUZZY LR LINEAR SYSTEMS
NEW MODELS AND ALGORITHMS FOR SOLUTIONS OF
SINGLESIGNED FULLY FUZZY LR LINEAR SYSTEMS
2
2
We present a model and propose an approach to compute an approximate solution of Fully Fuzzy Linear System $(FFLS)$ of equations in which all the components of the coefficient matrix are either nonnegative or nonpositive. First, in discussing an $FFLS$ with a nonnegative coefficient matrix, we consider an equivalent $FFLS$ by using an appropriate permutation to simplify fuzzy multiplications. To solve the $m times n$ permutated system, we convert it to three $m times n$ real linear systems, one being concerned with the cores and the other two being related to the left and right spreads. To decide whether the core system is consistent or not, we use the modified Huang algorithm of the class of $ABS$ methods.If the core system is inconsistent, an appropriate unconstrained least squares problem is solved for an approximate solution.The sign of each component of the solution is decided by the sign of its core. Also, to know whether the left and right spread systems are consistent or not, we apply the modified Huang algorithm again. Appropriate constrained least squares problems are solved, when the spread systems are inconsistent or do not satisfy fuzziness conditions.Then, we consider the $FFLS$ with a mixed singlesigned coefficient matrix, in which each component of the coefficient matrix is either nonnegative or nonpositive. In this case, we break the $m times n$ coefficient matrix up to two $m times n$ matrices, one having only nonnegative and the other having only nonpositive components, such that their sum yields the original coefficient matrix. Using the distributive law, we convert each $m times n$ $FFLS$ into two real linear systems where the first one is related to the cores with size $m times n$ and the other is $2m times 2n$ and is related to the spreads. Here, we also use the modified Huang algorithm to decide whether these systems are consistent or not. If the first system is inconsistent or the second system does not satisfy the fuzziness conditions, we find an approximate solution by solving a respective least squares problem. We summarize the proposed approach by presenting two computational algorithms. Finally, the algorithms are implemented and effectively tested by solving various randomly generated consistent as well as inconsistent numerical test problems.
1
We present a model and propose an approach to compute an approximate solution of Fully Fuzzy Linear System $(FFLS)$ of equations in which all the components of the coefficient matrix are either nonnegative or nonpositive. First, in discussing an $FFLS$ with a nonnegative coefficient matrix, we consider an equivalent $FFLS$ by using an appropriate permutation to simplify fuzzy multiplications. To solve the $m times n$ permutated system, we convert it to three $m times n$ real linear systems, one being concerned with the cores and the other two being related to the left and right spreads. To decide whether the core system is consistent or not, we use the modified Huang algorithm of the class of $ABS$ methods.If the core system is inconsistent, an appropriate unconstrained least squares problem is solved for an approximate solution.The sign of each component of the solution is decided by the sign of its core. Also, to know whether the left and right spread systems are consistent or not, we apply the modified Huang algorithm again. Appropriate constrained least squares problems are solved, when the spread systems are inconsistent or do not satisfy fuzziness conditions.Then, we consider the $FFLS$ with a mixed singlesigned coefficient matrix, in which each component of the coefficient matrix is either nonnegative or nonpositive. In this case, we break the $m times n$ coefficient matrix up to two $m times n$ matrices, one having only nonnegative and the other having only nonpositive components, such that their sum yields the original coefficient matrix. Using the distributive law, we convert each $m times n$ $FFLS$ into two real linear systems where the first one is related to the cores with size $m times n$ and the other is $2m times 2n$ and is related to the spreads. Here, we also use the modified Huang algorithm to decide whether these systems are consistent or not. If the first system is inconsistent or the second system does not satisfy the fuzziness conditions, we find an approximate solution by solving a respective least squares problem. We summarize the proposed approach by presenting two computational algorithms. Finally, the algorithms are implemented and effectively tested by solving various randomly generated consistent as well as inconsistent numerical test problems.
1
26
R.
Ezzati
R.
Ezzati
Department of Mathematics, Karaj Branch, Islamic Azad University,
31485  313, Karaj, Iran
Department of Mathematics, Karaj Branch,
Iran
ezati@kiau.ac.ir
S.
Khezerloo
S.
Khezerloo
Department of Mathematics, Karaj Branch, Islamic Azad University,
31485  313, Karaj, Iran
Department of Mathematics, Karaj Branch,
Iran
S khezerloo@yahoo.com
Z.
Valizadeh
Z.
Valizadeh
Department of Mathematics, Karaj Branch, Islamic Azad University,
31485  313, Karaj, Iran
Department of Mathematics, Karaj Branch,
Iran
z valizadeh@kiau.ac.ir
N. i
MahdaviAmiri
N.
MahdaviAmiri
Department of Mathematical Sciences, Sharif University of Tech
nology, 1458  889694, Tehran, Iran
Department of Mathematical Sciences, Sharif
Iran
nezamm@sina.sharif.edu
LR fuzzy numbers
Singlesigned fuzzy numbers
Fully fuzzy linear systems
ABS algorithms
Least squares problems
[[1] J. Abay, C. G. Broyden and E. Spedicato, A class of direct methods for linear systems,##Numerische Mathematik, 45 (1984), 361376.##[2] J. Abay and E. Spedicato, ABS projection alegorithms: mathematical techniques for linear##and nonlinear equations, John Wiley and Sons, 1989.##[3] S. Abbasbandy, A. Jafarian and R. Ezzati, Conjugate gradient method for fuzzy symmetric##positive denite system of linear equations, Applied Mathematics and Computation, 171##(2005), 11841191.##[4] S. Abbasbandy and M. Alavi, A method for solving fuzzy linear systems, Iranian Journal of##Fuzzy Systems, 2(2) (2005), 3743.##[5] S. Abbasbandy, B. Asady and M. Alavi, Fuzzy general linear systems, Applied Mathematics##and Computation, 169 (2005), 3440.##[6] T. Allahviranloo, Numerical methods for fuzzy system of linear equations, Applied Mathe##matics and Computation, 155 (2004), 493502. ##[7] T. Allahviranloo, Revised solution of an overdetermined fuzzy linear system of equations,##International Journal of Computational Cognition, 6(3) (2008), 6670.##[8] T. Allahviranloo, Successive over relaxation iterative method for fuzzy system of linear equa##tions, Applied Mathematics and Computation, 162 (2005), 189196.##[9] T. Allahviranloo, The Adomian decomposition method for fuzzy system of linear equations,##Applied Mathematics and Computation, 163 (2005), 553563.##[10] T. Allahviranloo, E. Ahmady, N. Ahmady and K. Shams Alketaby, Block Jacobi twostage##method with GaussSidel inner iterations for fuzzy system of linear equations, Applied Math##ematics and Computation, 175 (2006), 12171228.##[11] M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear programming: theory and algo##rithms, John Wiley and Sons, 2006.##[12] J. J. Buckley and Y. Qu, Solving system of linear fuzzy equations, Fuzzy Sets and Systems,##43 (1991), 3343.##[13] M. Dehghan and B. Hashemi, Iterative the solution of fuzzy linear systems, Applied Mathe##matics and Computation, 175 (2006), 645674.##[14] M. Dehghan and B. Hashemi, Solution of the fully fuzzy linear systems using the decompo##sition procedure, Applied Mathematics and Computation, 182 (2006), 15681580.##[15] M. Dehghan, B. Hashemi and M. Ghatee, Computational methods for solving fully fuzzy##linear systems, Applied Mathematics and Computation, 179 (2006), 328343.##[16] M. Dehghan, B. Hashemi and M. Ghatee, Solution of the fully fuzzy linear systems using##iterative techniques, Chaos Solutions and Fractals, 34 (2007), 316336.##[17] D. Dubois and H. Prade, Fuzzy sets and systems: theory and applications, Academic Press,##New York, 1980.##[18] H. Esmaeili, N. MahdaviAmiri and E. Spedicato, A class of ABS algorithms for Diophantine##linear systems, Numerische Mathematik, 90 (2001), 101115.##[19] R. Ezzati, Approximate symmetric least square solutions of general fuzzy linear systems,##Applied and Computational Mathematics, 9(2) (2010), 220225.##[20] R. Ezzati, Solving fuzzy linear systems, Soft Computing, 15 (2011), 193197.##[21] M. Friedman, M. Ming and A. Kandel, Fuzzy linear systems, Fuzzy Sets and Systems, 96##(1998), 201209.##[22] R. Ghanbari, N. MahdaviAmiri and R. Yousefpour, Exact and approximate solutions of fuzzy##LR linear systems: new algorithms using a least squares model and ABS approach, Iranian##Journal of Fuzzy Systems, 7(2) (2010), 118.##[23] M. S. Hashemi, M. K. Mirnia and S. Shahmorad, Solving fuzzy linear systems by using the##schur complement when cocent matrix is an Mmatrix, Iranian Journal of Fuzzy Systems,##15(3) (2008), 1529.##[24] A. Kauman and M. M. Gupta, Introduction to fuzzy arithmetic: theory and application,##Van Nostrand Reinhold, New York, 1991.##[25] M. Khorramizadeh and N. MahdaviAmiri, Integer extended ABS algorithms and possible##control of intermediate results for linear Diophantine systems, 4OR: A Quarterly Journal of##Operations Research, 7(2) (2009), 145167.##[26] M. Khorramizadeh and N. MahdaviAmiri, On solving linear Diophantine systems using##generalized Rosser's algorithm, Bulletin of Iranian Mathematical Society, 34(2) (2008), 1##[27] J. Nocedal and S. J. Wright, Numerical optimization, Springer, 2006.##[28] E. Spedicato, E. Bodon, A. Del Popolo and N. MahdaviAmiri, ABS methods and ABSPACK##for linear systems and optimization: a review, 4OR, 1 (2003), 5166.##[29] L. A. Zadeh, A fuzzysettheoretic interpretation of linguistic hedges, Journal of Cybernetics,##2 (1972), 434.##[30] L. A. Zadeh, The concept of the linguistic variable and its application to approximate rea##soning, Information Sciences, 8 (1975), 199249.##[31] H. J. Zimmermann, Fuzzy set theory and its applications, Kluwer Academic Press, Dordrecht,##]
MULTIOBJECTIVE OPTIMIZATION WITH PREEMPTIVE
PRIORITY SUBJECT TO FUZZY RELATION
EQUATION CONSTRAINTS
MULTIOBJECTIVE OPTIMIZATION WITH PREEMPTIVE
PRIORITY SUBJECT TO FUZZY RELATION
EQUATION CONSTRAINTS
2
2
This paper studies a new multiobjective fuzzy optimization prob lem. The objective function of this study has dierent levels. Therefore, a suitable optimized solution for this problem would be an optimized solution with preemptive priority. Since, the feasible domain is nonconvex; the tra ditional methods cannot be applied. We study this problem and determine some special structures related to the feasible domain, and using them some methods are proposed to reduce the size of the problem. Therefore, the prob lem is being transferred to a similar 01 integer programming and it may be solved by a branch and bound algorithm. After this step the problem changes to solve some consecutive optimized problem with linear objective function on discrete region. Finally, we give some examples to clarify the subject.
1
This paper studies a new multiobjective fuzzy optimization prob lem. The objective function of this study has dierent levels. Therefore, a suitable optimized solution for this problem would be an optimized solution with preemptive priority. Since, the feasible domain is nonconvex; the tra ditional methods cannot be applied. We study this problem and determine some special structures related to the feasible domain, and using them some methods are proposed to reduce the size of the problem. Therefore, the prob lem is being transferred to a similar 01 integer programming and it may be solved by a branch and bound algorithm. After this step the problem changes to solve some consecutive optimized problem with linear objective function on discrete region. Finally, we give some examples to clarify the subject.
27
45
Esmaile
Khorram
Esmaile
Khorram
Faculty of Mathematics and Computer Science, Amirkabir Uni
versity of Technology, 424,Hafez Ave.,15914,Tehran, Iran
Faculty of Mathematics and Computer Science,
Iran
eskhor@aut.ac.ir
Vahid
Nozari
Vahid
Nozari
Faculty of Mathematics and Computer Science, Amirkabir University
of Technology, 424,Hafez Ave.,15914,Tehran, Iran
Faculty of Mathematics and Computer Science,
Iran
vahid78mu@gmail.com
Fuzzy relation equation
Preemptive priority
Branch & Bound
Multi objective
Linear objective
Optimal solution
Binding variable
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ORDERED IDEAL INTUITIONISTIC FUZZY MODEL
OF FLOOD ALARM
ORDERED IDEAL INTUITIONISTIC FUZZY MODEL
OF FLOOD ALARM
2
2
An efficient flood alarm system may significantly improve public safety and mitigate damages caused by inundation. Flood forecasting is undoubtedly a challenging field of operational hydrology and a huge literature has been developed over the years. In this paper, we first define ordered ideal intuitionistic fuzzy sets and establish some results on them. Then, we define similarity measures between ordered ideal intuitionistic fuzzy sets (OIIFS) and apply these similarity measures to five selected sites of Kerala, India to predict potential flood.
1
An efficient flood alarm system may significantly improve public safety and mitigate damages caused by inundation. Flood forecasting is undoubtedly a challenging field of operational hydrology and a huge literature has been developed over the years. In this paper, we first define ordered ideal intuitionistic fuzzy sets and establish some results on them. Then, we define similarity measures between ordered ideal intuitionistic fuzzy sets (OIIFS) and apply these similarity measures to five selected sites of Kerala, India to predict potential flood.
47
60
Sunny Joseph
Kalayathankal
Sunny Joseph
Kalayathankal
Department of Mathematics, K.E.College, Mannanam,
Kottayam, 686561, Kerala, India
Department of Mathematics, K.E.College, Mannanam,
India
sunnyjose2000@yahoo.com
G.
Suresh Singh
G.
Suresh Singh
Department of Mathematics, University of Kerala, Trivandrum,
695581, Kerala, India
Department of Mathematics, University of
India
sureshsinghg@yahoo.co.in
Rainfall
Ordered intuitionistic fuzzy set
Flood
Simulation
[[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 8796.##[2] J. J. Buckley, K. D.Reilly and L. J. Jowers, Simulating continuous fuzzy systems, Iranian##Journal of Fuzzy Systems, 2(1) (2005), 118. ##[3] J. M. Cadenas and J. L. Verdegay, A primer on fuzzy optimization models and methods,##Iranian Journal of Fuzzy Systems, 3(1) (2006), 121.##[4] S. M. Chen, S. M Yeh and P. H. Hasiao, A comparison of similarity measures of fuzzy values,##Fuzzy Sets and Systems, 72 (1995), 7989.##[5] Dubois and Prade, Two fold fuzzy sets and rough sets  some issues in knowledge representation,##Fuzzy Sets and Systems, (1987), 318.##[6] J. A. Goguen, Lfuzzy sets, J. Math. Anal. Appl., 18 (1967), 145157.##[7] S. J. Kalayathankal, G. Suresh Singh, P. B. Vinodkumar, S. Joseph and J. Thomas, Ordered##intuitionistic fuzzy soft model of flood alarm, Iranian Journal of Fuzzy Systems, 8(1) (2011),##[8] S. J. Kalayathankal, G. Suresh Singh and P. B. Vinodkumar, OIIF model of flood alarm,##Global Journal of Mathematical Sciences: Theory and Practical, 1(1) (2009), 18.##[9] S. J. Kalayathankal and G. Suresh Singh, IFS model of flood alarm, Global Journal of Pure##and Applied Mathematics, 9 (2009), 1522.##[10] S. J. Kalayathankal and G. Suresh Singh, A fuzzy soft flood alarm model, Mathematics and##Computers in Simulation, 80 (2010), 887893.##[11] S. J. Kalayathankal, G. Suresh Singh and P. B. Vinodkumar, Ordered intuitionistic fuzzy soft##sets, Journal of Fuzzy Mathematics, 18(4) (2010), 991998.##[12] S. Li Chen, The application of comprehensive fuzzy judgement in the interpretation of waterflooded##reservoirs, The Journal of Fuzzy Mathematics, 9(3) (2001), 739743.##[13] I. Mahdavi, N. MahdaviAmiri, A. Heidarzade and R. Nourifar, Designing a model of fuzzy##TOPSIS in multiple criteria decision making, Applied Mathematics and Computation, 206##(2008), 607617.##[14] P. C. Nayak, K. P. Sudheer and K. S. Ramasastri, Fuzzy computing based rainfallrunoff##model for real time flood forecasting, Hydrological Processes, 19 (2005), 955968.##[15] W. Pedrycz, Distributed and collaborative fuzzy modeling, Iranian Journal of Fuzzy Systems,##4(1) (2007), 119.##[16] M. M. Xia and Z. S. Xu, Some new similarity measures for intuitionistic fuzzy values and##their application in group decision making, Journal of Systems Science and Systems Engineering,##19 (2010), 430452.##[17] Z. S. Xu and R. R. Yager, Dynamic intuitionistic fuzzy multiattribute decision making,##International Journal of Approximate Reasoning, 48 (2008), 246262.##[18] Z. S. Xu, On similarity measures of intervalvalued intuitionistic fuzzy sets and their application##to pattern recognitions, Journal of Southeast University, 23 (2007), 139143.##[19] Z. S. Xu and J. Chen, An overview of distance and similarity measures of intuitionistic##fuzzy sets, International Journal of Uncertainty, Fuzziness and KnowledgeBased Systems,##16 (2008), 529555.##[20] Z. S. Xu and R. R. Yager, Intuitionistic and intervalvalued intutionistic fuzzy preference##relations and their measures of similarity for the evaluation of agreement within a group,##Fuzzy Optimization and Decision Making, 8 (2009), 123139.##[21] Z. S. Xu, J. Chen and J. J. Wu, Clustering algorithm for intuitionistic fuzzy sets, Information##Sciences, 178 (2008), 37753790.##[22] Z. Xu, Some similarity measures of intuitionistic fuzzy sets and their applications to multiple##attribute decision making, Fuzzy Optim.Decis.Making, 6 (2007), 109121.##[23] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338353.##]
PRICING STOCKS BY USING FUZZY DIVIDEND
DISCOUNT MODELS
PRICING STOCKS BY USING FUZZY DIVIDEND
DISCOUNT MODELS
2
2
Although the classical dividend discount model (DDM) is a wellknown and widely used model in evaluating the intrinsic price of common stock, the practical pattern of dividends, required rate of return or growth rate of dividend do not generally coincide with any of the model’s assumptions. It is just the opportunity to develop a fuzzy logic system that takes these vague parameters into account. This paper extends the classical DDMs to more realistic fuzzy pricing models in which the inherent imprecise information will be fuzzified as triangular fuzzy numbers, and introduces a novel signed distance method to defuzzify these fuzzy parameters without considering the membership functions. Through the conscientious mathematical derivation, the fuzzy dividend discount models (FDDMs) proposed in this paper can be regarded as one more explicit extension of the classical (crisp) DDMs, so that stockholders can use it to make a specific analysis and insight into the intrinsic value of stock.
1
Although the classical dividend discount model (DDM) is a wellknown and widely used model in evaluating the intrinsic price of common stock, the practical pattern of dividends, required rate of return or growth rate of dividend do not generally coincide with any of the model’s assumptions. It is just the opportunity to develop a fuzzy logic system that takes these vague parameters into account. This paper extends the classical DDMs to more realistic fuzzy pricing models in which the inherent imprecise information will be fuzzified as triangular fuzzy numbers, and introduces a novel signed distance method to defuzzify these fuzzy parameters without considering the membership functions. Through the conscientious mathematical derivation, the fuzzy dividend discount models (FDDMs) proposed in this paper can be regarded as one more explicit extension of the classical (crisp) DDMs, so that stockholders can use it to make a specific analysis and insight into the intrinsic value of stock.
61
78
HueiWen
Lin
HueiWen
Lin
Department of Finance and Banking, Aletheia University, 32 ChenLi
Street, 25103, New Taipei City, Taiwan (R.O.C.)
Department of Finance and Banking, Aletheia
Taiwan
au4345@mail.au.edu.tw
JingShing
Yao
JingShing
Yao
Department of Mathematics, National Taiwan University, No.1, Sec.
4, Roosevelt Rd., Taipei City 106, Taiwan (R.O.C.)
Department of Mathematics, National Taiwan
Taiwan
hflu.chibi@msa.hinet.net
Fuzzy set
Pricing stock
Dividend discount model (DDM)
$l$signed distance method
Uniform convergence
[[1] L. Brand, Advance calculus: an introduction to classical analysis, New York, 1955.##[2] E. F. Brigham, Fundamentals of nancial management, The Dryden Press, New York, 1992.##[3] J. J. Buckley, The fuzzy mathematics of nance, Fuzzy Sets and Systems, 21 (1987), 257273.##[4] H. Dourra and P. Siy, Investment using technical analysis and fuzzy logic, Fuzzy Sets and##Systems, 127 (2002), 221240.##[5] M. J. Gordon, The investment, nancing, and valuation of the corporation, Homewood,##Illinois: Richard D. Irwin, 1962.##[6] W. J. Hurley and L. D. Johnson, A realistic dividend valuation model, Financial Analysts##Journal, JulyAugust(1994), 5054.##[7] A. Kaufmann and M. M. Gupta, Introduction to fuzzy arithmetic: theory and applications,##Van Nostrand Reinhold, New York, 1991.##[8] D. Kuchta, Fuzzy capital budgeting, Fuzzy Sets and Systems, 111 (2000), 367385.##[9] R. J. Kuo, C. H. Chen and Y. C. Hwang, An intelligent stock trading decision support##system through integration of genetic algorithm based fuzzy neural network and articial##neural network, Fuzzy Sets and Systems, 118 (2001), 2145.##[10] M. L. Leibowitz and S. Kogelman, The growth illusion: the P/E 'cost' of earnings growth,##Financial Analysts journal, MarchApril(1994), 3648.##[11] M. Li Calzi, Towards a general setting for the fuzzy mathematics of nance, Fuzzy Sets and##Systems, 35 (1990), 265280.##[12] P. Liang and F. Song, Computeraided risk evaluation system for capital investment, Omega,##22(4) (1994), 391400.##[13] A. Maturo, On some structures of fuzzy numbers, Iranian Journal of Fuzzy Systems, 6(4)##(2009), 4959.##[14] P. M . Pu and Y. M . Liu, Fuzzy topology 1, neighborhood structure of a fuzzy point and##Mooresmith convergence, Journal of Mathematical Analysis and Applications, 76 (1980),##[15] W. F. Sharpe, G. J . Alexander and J. V. Bailey, Investments, Englewood Cliffs PrenticeHall##Inc., 1999.##[16] R. J. Shiller, Do stock prices move too much to be justied by subsequent change in dividends?##American Economic Review, 71 (1981), 421436.##[17] A. K. Shyamal and M. Pal, Triangular fuzzy matrices, Iranian Journal of Fuzzy Systems,##4(1) (2007), 7587.##[18] R. Viertl and D. Hareter, Fuzzy information and stochastics, Iranian Journal of Fuzzy Systems,##1(1) (2004), 4356.##[19] Y. F. Wang, Predicting stock price using fuzzy grey prediction system, Expert Systems with##Applications, 22 (2002), 3339.##[20] Y. F. Wang, Mining stock price using fuzzy rough set system, Expert Systems with Applications,##24 (2003), 1323.##[21] J. S. Yao and K. M. Wu, Ranking fuzzy numbers based on decomposition principle and signed##distance, Fuzzy Sets and Systems, 116 (2000), 275288.##[22] R. Zhao and R. Govind, Defuzzication of fuzzy intervals, Fuzzy Sets and Systems, 43 (1991),##]
ROUGH SET OVER DUALUNIVERSES IN FUZZY
APPROXIMATION SPACE
ROUGH SET OVER DUALUNIVERSES IN FUZZY
APPROXIMATION SPACE
2
2
To tackle the problem with inexact, uncertainty and vague knowl edge, constructive method is utilized to formulate lower and upper approx imation sets. Rough set model over dualuniverses in fuzzy approximation space is constructed. In this paper, we introduce the concept of rough set over dualuniverses in fuzzy approximation space by means of cut set. Then, we discuss properties of rough set over dualuniverses in fuzzy approximation space from two viewpoints: approximation operators and cut set of fuzzy set. Reduction of attributes and rules extraction of rough set over dualuniverses in fuzzy approximation space are presented. Finally, an example of disease diagnoses expert system illustrates the possibility and eciency of rough set over dualuniverses in fuzzy approximation space.
1
To tackle the problem with inexact, uncertainty and vague knowl edge, constructive method is utilized to formulate lower and upper approx imation sets. Rough set model over dualuniverses in fuzzy approximation space is constructed. In this paper, we introduce the concept of rough set over dualuniverses in fuzzy approximation space by means of cut set. Then, we discuss properties of rough set over dualuniverses in fuzzy approximation space from two viewpoints: approximation operators and cut set of fuzzy set. Reduction of attributes and rules extraction of rough set over dualuniverses in fuzzy approximation space are presented. Finally, an example of disease diagnoses expert system illustrates the possibility and eciency of rough set over dualuniverses in fuzzy approximation space.
79
91
Ruixia
Yan
Ruixia
Yan
School of Management, Shanghai University of Engineering Science,
Shanghai 201620, P. R. China and Glorious Sun School of Business Administration,
Donghua Universty, Shanghai 200051, P. R.China
School of Management, Shanghai University
China
yanruixia@gmail.com
Jianguo
Zheng
Jianguo
Zheng
Glorious Sun School of Business Administration, Donghua Univer
sity, Shanghai 200051, P. R.China
Glorious Sun School of Business Administration,
China
zjg@dhu.edu.cn
Jinliang
Liu
Jinliang
Liu
Department of Applied Mathematics, Nanjing University of Finance
and Economics, Nanjing, 210046, P.R.China
Department of Applied Mathematics, Nanjing
China
liujinliang@vip.163.com
Chaoyong
Qin
Chaoyong
Qin
College of Mathematics and Information Sciences of Guangxi Univer
sity, Naning 530004, P. R. China and Glorious Sun School of Business Administration,
Donghua Universty, Shanghai 200051, P. R.China
College of Mathematics and Information Sciences
China
qcy@dhu.edu.cn
[[1] D. Dubois and H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of##General Systems, 17(23) (1990), 191209.##[2] V. Fotea, Fuzzy rough nary subhypergroups, Iranian Journal of Fuzzy Systems, 5(3) (2008),##[3] S. Greco, B. Matarazzo and R. Slowinski, Rough approximation of a preference relation by##dominance relations, European Journal of Operational Research, 117(1) (1999), 6383.##[4] T. Li, Rough approximation operators on two universes of discourse and their fuzzy exten##sions, Fuzzy Sets and Systems, 159(22) (2008), 30333050.##[5] T. Li and W. Zhang, Rough fuzzy approximations on two universes of discourse, Information##Sciences, 178(3) (2008), 892906.##[6] G. Liu, Rough set theory based on two universal sets and its applications, KnowledgeBased##Systems, 2009.##[7] G. Liu and W. Zhu, The algebraic structures of generalized rough set theory, Information##Sciences, 178(21) (2008), 41054113.##[8] N. Mac Parthalain and Q. Shen, Exploring the boundary region of tolerance rough sets for##feature selection, Pattern Recognition, 42(5) (2009), 655667.##[9] J. Mi, Y. Leung, H. Zhao and T. Feng, Generalized fuzzy rough sets determined by a triangular##norm, Information Sciences, 178(16) (2008), 32033213.##[10] Z. Pawlak, Rough sets, International Journal of Computer and Information Science, 11##(1982), 341356.##[11] Z. Pawlak, Rough sets: theoretical aspects of reasoning about data, Springer, 1991.##[12] D. Pei, A generalized model of fuzzy rough sets, International Journal of General Systems,##34(5) (2005), 603613.##[13] D. Pei and Z. Xu, Rough set models on two universes, International Journal of General##Systems, 33(5) (2004), 569581.##[14] S. Roman and V. Daniel, A genaralized denition of rough approximations based on similar##ity, IEEE Transactions on Knowledge and Data Engineering, 12 (2000), 331336.##[15] Y. Shen and F. Wang, Variable precision rough set model over two universes and its prop##erties, Soft ComputingA Fusion of Foundations, Methodologies and Applications, 15(3)##(2011), 557567.##[16] A. Skowron and J. Stepaniuk, Tolerance approximation spaces, Fundamenta Informaticae,##27(2/3) (1996), 245253.##[17] B. Sun and W. Ma, Fuzzy rough set model on two dierent universes and its application,##Applied Mathematical Modelling, 2010.##[18] S. Wong, L. Wang and Y. Yao, On modeling uncertainty with interval structures, Computa##tional Intelligence, 11(2) (1995), 406426.##[19] W. Wu, Y. Leung and J. Mi, On characterizations offuzzy rough approximation operators,##Fuzzy Sets and Systems, 154(1) (2005), 76102.##[20] W.Wu, Y. Leung and W. Zhang, On generalized rough fuzzy approximation operators, Trans##actions on Rough Sets V, (2006), 263284.##[21] Q. Wu and Z. Liu, Real formal concept analysis based on greyrough set theory, Knowledge##Based Systems, 22(1) (2009), 3845. ##[22] W. Wu, J. Mi and W. Zhang, Generalized fuzzy rough sets, Information Sciences, 151 (2003),##[23] W. Wu and W. Zhang, Constructive and axiomatic approaches of fuzzy approximation oper##ators, Information Sciences, 159(34) (2004), 233254.##[24] J. Z. R. Yan and J. Liu, Rough set over dualuniverses and its applications in: expert systems,##ICIC Express Letters, 4(3) (2010), 833838.##[25] R. Yan, J. Zheng, J. Liu and Y. Zhai, Research on the model of rough set over dualuniverses,##KnowledgeBased Systems, 23(8) (2010), 817822.##[26] Y. Yao, Probabilistic rough set approximations, International Journal of Approximate Rea##soning, 49(2) (2008), 255271.##[27] Y. Yao, Generalized rough set models, Rough Sets in Knowledge Discovery, 1 (1998), 286318.##[28] Y. Yao and T. Lin, Generalization of rough sets using modal logic, Intelligent Automation##and Soft Computing, 2(2) (1996), 103120.##[29] L. Zadeh, Fuzzy sets, Information and control, 8(3) (1965), 338353.##[30] K. Zaras, Rough approximation of a preference relation by a multiattribute dominance for##deterministic, stochastic and fuzzy decision problems, European Journal of Operational Re##search, 159(1) (2004), 196206.##[31] H. Zhang, W. Zhang and W. Wu, On characterization of generalized intervalvalued fuzzy##rough sets on two universes of discourse, International Journal of Approximate Reasoning,##51(1) (2009), 5670.##[32] W. Ziarko, Variable precision rough set model, Journal of Computer and System Sciences,##46(1) (1993), 3959.##]
A BIOBJECTIVE PROGRAMMING APPROACH TO SOLVE
MATRIX GAMES WITH PAYOFFS OF ATANASSOV’S
TRIANGULAR INTUITIONISTIC FUZZY NUMBERS
A BIOBJECTIVE PROGRAMMING APPROACH TO SOLVE
MATRIX GAMES WITH PAYOFFS OF ATANASSOV’S
TRIANGULAR INTUITIONISTIC FUZZY NUMBERS
2
2
The intuitionistic fuzzy set has been applied to game theory very rarely since it was introduced by Atanassov in 1983. The aim of this paper is to develop an effective methodology for solving matrix games with payoffs of Atanassov’s triangular intuitionistic fuzzy numbers (TIFNs). In this methodology, the concepts and ranking order relations of Atanassov’s TIFNs are defined. A pair of biobjective linear programming models for matrix games with payoffs of Atanassov’s TIFNs is derived from two auxiliary Atanassov’s intuitionistic fuzzy programming models based on the ranking order relations of Atanassov’s TIFNs defined in this paper. An effective methodology based on the weighted average method is developed to determine optimal strategies for two players. The proposed method in this paper is illustrated with a numerical example of the market share competition problem.
1
The intuitionistic fuzzy set has been applied to game theory very rarely since it was introduced by Atanassov in 1983. The aim of this paper is to develop an effective methodology for solving matrix games with payoffs of Atanassov’s triangular intuitionistic fuzzy numbers (TIFNs). In this methodology, the concepts and ranking order relations of Atanassov’s TIFNs are defined. A pair of biobjective linear programming models for matrix games with payoffs of Atanassov’s TIFNs is derived from two auxiliary Atanassov’s intuitionistic fuzzy programming models based on the ranking order relations of Atanassov’s TIFNs defined in this paper. An effective methodology based on the weighted average method is developed to determine optimal strategies for two players. The proposed method in this paper is illustrated with a numerical example of the market share competition problem.
93
110
DengFeng
Li
DengFeng
Li
School of Management, Fuzhou University, No. 2, Xueyuan Road,
Daxue New District, Fuzhou 350108, Fujian, China
School of Management, Fuzhou University,
China
lidengfeng@fzu.edu.cn, dengfengli@sina.com
JiangXia
Nan
JiangXia
Nan
School of Mathematics and Computing Sciences, Guilin University
of Electronic Technology, Guilin, Guangxi 541004, China
School of Mathematics and Computing Sciences,
China
nanjiangxia@guet.edu.cn
ZhenPeng
Tang
ZhenPeng
Tang
School of Management, Fuzhou University, No. 2, Xueyuan Road,
Daxue New District, Fuzhou 350108, Fujian, China
School of Management, Fuzhou University,
China
zhenpt@126.com
KeJia
Chen
KeJia
Chen
School of Management, Fuzhou University, No. 2, Xueyuan Road,
Daxue New District, Fuzhou 350108, Fujian, China
School of Management, Fuzhou University,
China
kjchen@fzu.edu.cn
XiaoDong
Xiang
XiaoDong
Xiang
School of Management, Fuzhou University, No. 2, Xueyuan Road,
Daxue New District, Fuzhou 350108, Fujian, China
School of Management, Fuzhou University,
China
xiangxiaodong2@yahoo.com.cn
FangXuan
Hong
FangXuan
Hong
School of Management, Fuzhou University, No. 2, Xueyuan Road,
Daxue New District, Fuzhou 350108, Fujian, China
School of Management, Fuzhou University,
China
hongfangxuan2@163.com
Uncertainty
Fuzzy set
Atanassov’s intuitionistic fuzzy set
Fuzzy number
Matrix game
Mathematical programming
[[1] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 8796.##[2] K. T. Atanassov, Intuitionistic fuzzy sets, SpringerVerlag, Heidelberg, Germany, 1999.##[3] K. T. Atanassov and G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy Sets and##Systems, 31 (1989), 343349.##[4] K. T. Atanassov, Answer to D. Dubois, S. Gottwald, P. Hajek, J. Kacprzyk and H. Prade’s##paper Terminological difficulties in fuzzy set theory  the case of ”intuitionistic fuzzy sets”,##Fuzzy Sets and Systems, 156 (2005), 496499.##[5] C. R. Bector and S. Chandra, Fuzzy mathematical programming and fuzzy matrix games,##Springer Verlag, Berlin, Germany, 2005.##[6] C. R. Bector, S. Chandra and V. Vijay, Matrix games with fuzzy goals and fuzzy linear##programming duality, Fuzzy Optimization and Decision Making, 3 (2004), 255269.##[7] C. R. Bector, S. Chandra and V. Vijay, Duality in linear programming with fuzzy parameters##and matrix games with fuzzy payoffs, Fuzzy Sets and Systems, 46(2) (2004), 253269.##[8] R. A. Borzooei and Y. B. Jun, Intuitionistic fuzzy hyper bckideals of hyper bckal gebras,##Iranian Journal of Fuzzy Systems, 1(1) (2004), 6578.##[9] P. Burillo and H. Bustince, Vague sets are intuitionistic fuzzy sets, Fuzzy Sets and Systems,##79 (1996), 403405.##[10] L. Campos, Fuzzy linear programming models to solve fuzzy matrix games, Fuzzy Sets and##Systems, 32 (1989), 275289.##[11] L. Campos and A. Gonzalez, Fuzzy matrix games considering the criteria of the players,##Kybernetes, 20 (1991), 1723.##[12] L. Campos, A. Gonzalez and M. A. Vila, On the use of the ranking function approach to##solve fuzzy matrix games in a direct way, Fuzzy Sets and Systems, 49 (1992), 193203.##[13] S. K. De, R. Biswas and A. R. Roy, An application of intuitionistic fuzzy sets in medical##diagnosis, Fuzzy Sets and Systems, 117 (2001), 209213.##[14] G. Deschrijver, Arithmetic operators in intervalvalued fuzzy set theory, Information Sciences,##177(14) (2007), 29062924.##[15] G. Deschrijver and E. E. Kerre, On the relationship between some extensions of fuzzy set##theory, Fuzzy Sets and Systems, 133 (2003), 227235.##[16] G. Deschrijver and E. E. Kerre, On the position of intuitionistic fuzzy set theory in the##framework of theories modeling imprecision, Information Sciences, 177 (2007), 18601866.##[17] D. Dubois, S. Gottwald, P. Hajek, J. Kacprzyk and H. Prade, Terminological difficulties in##fuzzy set theorythe case of ”Intuitionistic Fuzzy Sets”, Fuzzy Sets and Systems, 156 (3)##(2005), 485491.##[18] D. Dubois and H. Prade, fuzzy sets and systems: theory and applications, Mathematics in##Science and Engineering, Academic Press, Berlin, Germany, 144 (1980).##[19] J. G. Garc and S. E. Rodabaugh, Ordertheoretic, topological, categorical redundancies of##intervalvalued sets, grey sets, vague sets, intervalvalued ”intuitionistic” sets, ”intuitionistic”##fuzzy sets and topologies, Fuzzy Sets and Systems, 156(3) (2005), 445484. ##[20] W. L. Gau and D. J. Buehrer, Vague sets, IEEE Transaction on Systems, Man, and Cybernetics,##23 (1993), 610614.##[21] J. Goguen, Lfuzzy sets, Journal of Mathematical Analysis and Applications, 18 (1967),##[22] E. E. Kerre and J. N. Mordeson, A historical overview of fuzzy mathematicas, New Mathematics##and Natural Computation, 1(1) (2005), 126.##[23] A. Khan, Y. B. Jun and M. Shabir, Ordered semigroups characterized by their intuitionistic##fuzzy biideals, Iranian Journal of Fuzzy Systems, 7(2) (2010), 5569.##[24] D. F. Li, Fuzzy constrained matrix games with fuzzy payoffs, The Journal of Fuzzy Mathematics,##7(4) (1999), 873880.##[25] D. F. Li, A fuzzy multiobjective programming approach to solve fuzzy matrix games, The##Journal of Fuzzy Mathematics, 7(4) (1999), 907912.##[26] D. F. Li, Multiattribute decision making models and methods using intuitionistic fuzzy sets,##Journal of Computer and System Sciences, 70 (2005), 7385.##[27] D. F. Li, A note on using intuitionistic fuzzy sets for faulttree analysis on printed circuit##board assembly, Microelectronics Reliability, 48 (2008), 1741.##[28] D. F Li, A ratio ranking method of triangular intuitionistic fuzzy numbers and its application##to MADM problems, Computers and Mathematics with Applications, 60(6) (2010), 1557##[29] D. F. Li, Representation of level sets and extension principles for Atanassov’s intuitionistic##fuzzy sets and algebraic operations, Critical View, 4 (2010), 6374.##[30] D. F. Li and C. T. Cheng, New similarity measures of intuitionistic fuzzy sets and application##to pattern recognitions, Pattern Recognition Letters, 23 (2002), 221225.##[31] D. F. Li and J. X. Nan, A nonlinear programming approach to matrix games with payoffs##of Atanassov’s intuitionistic fuzzy sets, International Journal of Uncertainty, Fuzziness and##KnowledgeBased Systems, 17(4) (2009), 585607.##[32] S. T. Liu and C. Kao, Solution of fuzzy matrix games: an application of the extension##principle, International Journal of Intelligent Systems, 22 (2007), 891903.##[33] A. Maturo, On some structures of fuzzy numbers, Iranian Journal of Fuzzy Systems, 6(4)##(2009), 4959.##[34] P. K. Nayak and M. Pal, Bimatrix games with intuitionistic fuzzy goals, Iranian Journal of##Fuzzy Systems, 7(1) (2010), 6579.##[35] I. Nishizaki and M. Sakawa, Equilibrium solutions in multiobjective bimatrix games with##fuzzy payoffs and fuzzy goals, Fuzzy Sets and Systems, 111(1) (2000), 99116.##[36] I. Nishizaki and M. Sakawa, Solutions based on fuzzy goals in fuzzy linear programming##games, Fuzzy Sets and Systems, 115(1) (2000), 105119.##[37] I. Nishizaki and M. Sakawa, Fuzzy and multiobjective games for conflict resolution, Physica##Verlag, Springer Verlag Company, Berlin, Germany, 2001.##[38] E. Pasha, A. Saiedifar and B. Asady, The percentiles of fuzzy numbers and their applications,##Iranian Journal of Fuzzy Systems, 6(1) (2009), 2744.##[39] M. Sakawa and I. Nishizaki, A lexicographical solution concept in an nperson cooperative##fuzzy game, Fuzzy Sets and Systems, 61 (1994), 265275.##[40] M. Sakawa and I. Nishizaki, Maxmin solutions for fuzzy multiobjective matrix games, Fuzzy##Sets and Systems, 67 (1994), 5369.##[41] E. Savas, (A)Double sequence spaces of fuzzy numbers via orlicz function, Iranian Journal##of Fuzzy Systems, 8(2) (2011), 91103.##[42] E. Szmidt and F. Baldwin, Intuitionistic fuzzy set functions, mass assignment theory, possibility##theory and histograms, 2006 IEEE International Conference on Fuzzy Systems Sheraton##Vancouver Wall Centre Hotel, Vancouver, BC, Canada, July 1621, (2006), 3541.##[43] G. Takeuti and S. Titani, Intuitionistic fuzzy logic and intuitionistic fuzzy set theory, The##Journal of Symbolic Logic, 49(3) (1984), 851866.##[44] V. Vijay, S. Chandra and C. R. Bector, Matrix games with fuzzy goals and fuzzy payoffs,##Omega, 33 (2005), 425429. ##[45] Z. S. Xu, Models for multiple attribute decision making with intuitionistic fuzzy information,##International Journal of Uncertainty, Fuzziness and KnowledgeBased Systems, 15 (2007),##[46] Z. S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems,##15 (2007), 11791187.##[47] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338356.##[48] L. A. Zadeh, Outline of a new approach to the analysis of complex systems and decision##processes intervalvalued fuzzy sets, IEEE Transactions on Systems, Man, and Cybernetics,##3 (1973), 2844.##]
(T) FUZZY INTEGRAL OF MULTIDIMENSIONAL FUNCTION
WITH RESPECT TO MULTIVALUED MEASURE
(T) FUZZY INTEGRAL OF MULTIDIMENSIONAL FUNCTION
WITH RESPECT TO MULTIVALUED MEASURE
2
2
Introducing more types of integrals will provide more choices todeal with various types of objectives and components in real problems. Firstly,in this paper, a (T) fuzzy integral, in which the integrand, the measure andthe integration result are all multivalued, is presented with the introductionof Tnorm and Tconorm. Then, some classical results of the integral areobtained based on the properties of Tnorm and Tconorm mainly. The presented integral can act as an aggregation tool which is especially useful inmany information fusing and data mining problems such as classication andprogramming.
1
Introducing more types of integrals will provide more choices todeal with various types of objectives and components in real problems. Firstly,in this paper, a (T) fuzzy integral, in which the integrand, the measure andthe integration result are all multivalued, is presented with the introductionof Tnorm and Tconorm. Then, some classical results of the integral areobtained based on the properties of Tnorm and Tconorm mainly. The presented integral can act as an aggregation tool which is especially useful inmany information fusing and data mining problems such as classication andprogramming.
111
126
Wanli
Liu
Wanli
Liu
Department of Spatial Informatics, China University of Mining and
Technology, Xuzhou, Jiangsu 221116, P. R. China
Department of Spatial Informatics, China
China
liuliucumt@126.com
Xiaoqiu
Song
Xiaoqiu
Song
Department of Mathematics, China University of Mining and Tech
nology, Xuzhou, Jiangsu 221116, P. R. China
Department of Mathematics, China University
China
songxiaoqiu@cumt.edu.cn
Qiuzhao
Zhang
Qiuzhao
Zhang
Department of Spatial Informatics, China University of Mining and
Technology, Xuzhou, Jiangsu 221116, P. R. China
Department of Spatial Informatics, China
China
qiuzhaocumt@163.com
Shubi
Zhang
Shubi
Zhang
Department of Spatial Informatics, China University of Mining and
Technology, Xuzhou, Jiangsu 221116, P. R. China
Department of Spatial Informatics, China
China
zhangsbi@vip.sina.com
$mathfrak{T}$norm
$mathfrak{T}$conorm
Multidimensional function
Multivalued measure
$(T)$ fuzzy integral
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PREDICTING URBAN TRIP GENERATION USING A FUZZY
EXPERT SYSTEM
PREDICTING URBAN TRIP GENERATION USING A FUZZY
EXPERT SYSTEM
2
2
One of the most important stages in the urban transportation planning procedure is predicting the rate of trips generated by each trac zone. Currently, multiple linear regression models are frequently used as a prediction tool. This method predicts the number of trips produced from, or attracted to each trac zone according to the values of independent variables for that zone. One of the main limitations of this method is its huge dependency on the exact prediction of independent variables in future (horizon of the plan). The other limitation is its many assumptions, which raise challenging questions of its application. The current paper attempts to use fuzzy logic and its capabilities to estimate the trip generation of urban zones. A fuzzy expert system is introduced, which is able to make suitable predictions using uncertain and inexact data. Results of the study on the data for Mashhad (Lon: 59.37 E, Lat: 36.19 N) show that this method can be a good competitor for multiple linear regression method, specially, when there is no exact data for independent variables.
1
One of the most important stages in the urban transportation planning procedure is predicting the rate of trips generated by each trac zone. Currently, multiple linear regression models are frequently used as a prediction tool. This method predicts the number of trips produced from, or attracted to each trac zone according to the values of independent variables for that zone. One of the main limitations of this method is its huge dependency on the exact prediction of independent variables in future (horizon of the plan). The other limitation is its many assumptions, which raise challenging questions of its application. The current paper attempts to use fuzzy logic and its capabilities to estimate the trip generation of urban zones. A fuzzy expert system is introduced, which is able to make suitable predictions using uncertain and inexact data. Results of the study on the data for Mashhad (Lon: 59.37 E, Lat: 36.19 N) show that this method can be a good competitor for multiple linear regression method, specially, when there is no exact data for independent variables.
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146
Amir Abbas i
Rassafi
Amir Abbas
Rassafi
Faculty of Engineering, Imam Khomeini International Univer
sity, Qazvin, 34149, Iran
Faculty of Engineering, Imam Khomeini Internationa
Iran
rasafi@ikiu.ac.ir
Roohollah
Rezaei
Roohollah
Rezaei
Faculty of Engineering, Imam Khomeini International University,
Qazvin, 34149, Iran
Faculty of Engineering, Imam Khomeini Internationa
Iran
te rezaei@yahoo.com
Mehdi
Hajizamani
Mehdi
Hajizamani
MITPortugal Program, Instituto Superior Tcnico, Technical
University of Lisbon, Lisbon, Portugal
MITPortugal Program, Instituto Superior
Portugal
mhajizamani@yahoo.com
Trip generation
Multiple linear regression
Membership function
Fuzzy rules
Fuzzy expert system
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