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A CONSTRAINED SOLID TSP IN FUZZY ENVIRONMENT:
TWO HEURISTIC APPROACHES
A CONSTRAINED SOLID TSP IN FUZZY ENVIRONMENT:
TWO HEURISTIC APPROACHES
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A solid travelling salesman problem (STSP) is a travelling salesman problem (TSP) where the salesman visits all the cities only once in his tour using dierent conveyances to travel from one city to another. Costs and environmental eect factors for travelling between the cities using dierent conveyances are dierent. Goal of the problem is to nd a complete tour with minimum cost that damages the environment least. An ant colony optimization (ACO) algorithm is developed to solve the problem. Performance of the algorithm for the problem is compared with another soft computing algorithm, Genetic Algorithm(GA). Problems are solved with crisp as well as fuzzy costs. For fuzzy cost and environmental eect factors, cost function as well as environment constraints become fuzzy. As optimization of a fuzzy objective function is not well de ned, fuzzy possibility approach is used to get optimal decision. To test the eciency of the algorithm, the problem is solved considering only one conveyance facility ignoring the environmental eect constraint, i.e., a classical two dimensional TSP (taking standard data sets from TSPLIB for solving the problem). Dierent numerical examples are used for illustration.
1
A solid travelling salesman problem (STSP) is a travelling salesman problem (TSP) where the salesman visits all the cities only once in his tour using dierent conveyances to travel from one city to another. Costs and environmental eect factors for travelling between the cities using dierent conveyances are dierent. Goal of the problem is to nd a complete tour with minimum cost that damages the environment least. An ant colony optimization (ACO) algorithm is developed to solve the problem. Performance of the algorithm for the problem is compared with another soft computing algorithm, Genetic Algorithm(GA). Problems are solved with crisp as well as fuzzy costs. For fuzzy cost and environmental eect factors, cost function as well as environment constraints become fuzzy. As optimization of a fuzzy objective function is not well de ned, fuzzy possibility approach is used to get optimal decision. To test the eciency of the algorithm, the problem is solved considering only one conveyance facility ignoring the environmental eect constraint, i.e., a classical two dimensional TSP (taking standard data sets from TSPLIB for solving the problem). Dierent numerical examples are used for illustration.
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28
Chiranjit
Changdar
Chiranjit
Changdar
Department of Computer Science, Raja N.L. Khan Women's
College, Midnapore, Paschim Medinipur, West Bengal, India721102
Department of Computer Science, Raja N.L.
India
chiranjit changdar@yahoo.co.in
Manas Kumar
Maiti
Manas Kumar
Maiti
Department of Mathematics, Mahishadal Raj College, Mahishadal,
Purba Medinipur, West Bengal, India721628
Department of Mathematics, Mahishadal Raj
India
manasmaiti@yahoo.co.in
Manoranjan
Maiti
Manoranjan
Maiti
Department of Mathematics, Vidyasagar University, Midnapore,
Paschim Medinipur, West Bengal, India721102
Department of Mathematics, Vidyasagar University,
India
mmaiti2005@yahoo.co.in
Solid travelling salesman problem
Fuzzy possibility
Ant colony optimization
Genetic Algorithm
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Johnson,Solution of largescale travelling salesman problem, Operations Research, 2 (1954), 393410. ##[7] M. Dorigo and L. M. Gambardella,Ant colony system: an cooperative learning approach to the travelling salesman problem, IEEE Transactions on Evolutionary Computation,1(1)(1997). ##[8] M. Dorigo and T. Stutzle,Ant colony optimization, prentice hall of India private limitde,New Delhi, 2006 ##[9] D. Dubois and H. Prade,Fuzzy sets and system  theory and application, Academic, NewYork, 1980. ##[10] A. P. Engelbrech,Fundamentals of computational swarm intelligence, Wiley, 2005. ##[11] O. Ergan and J. B. Orlin,A dynamic programming methodology in very large sccale neighbourhood applied to travelling Salesman problem, Discrete Optimization, 3 (2006), 7885. ##[12] F. Focacci, A. Lodi, M. Milano ,A hybrid exact algorithm for the TSPTW, INFORM Journal on Computing,14(4) (2002),403417. ##[13] D. E. Goldberg,Genetic algorithms: search, optimization and machine learning, Addison Wesley, assachusetts, 1989. ##[14] T. Ibaraki, S. Imahori, M. Kubo, T. Masuda, T. Uno and M. Yagiura,fective local search algorithm for routing and scheduling problems with general time window constraints, Transportation Science,39(2)(2005), 206232. ##[15] J. Knox,The application of Tabu search to the symmetric traveling salesman problem, h.D.Dissertation, University of Colorado, 1989. ##[16] A. Kumar, A. Gupta and M. K. Sharma,Application of Tabu search for solving the biobjective warehouse problem in a fuzzy environment, Iranian Journal of Fuzzy Systems, 9(1)(2012), 119. ##[17] E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan and D. B. Shmoys,The traveling salesman problem: G. E. Re Guided tour of combinatorial optimization, Wiley and Sons, New York,1985. ##[18] S. Lin and B. W. Kernighan,An effective heuristic algorithm for the traveling salesman problem, erations Research, 21 (1973), 498516. ##[19] Y. Liu ,Dierent initial solution generators in genetic algorithms for solving the probabilistic traveling salesman problem, Applied Mathematics and Computation, 216 (2010), 125137. ##[20] I. Mahdavi, N. MadhaviAmiri AND S. Nejati,Algorithms for biobjective shortest path problems in fuzzy networks, Iranian Journal of Fuzzy Systems, 8(4) (2011), 737. ##[21] M. K. Maiti and M. Maiti,Twostorage inventory model with lotsize dependent fuzzy leadtime under possibility constraints via genetic algorithm, European Journal of Operational Research,179 (2007), 352371. ##[22] M. K. Maiti and M. Maiti,Fuzzy inventory model with two warehouses under possibility constraints, Fuzzy Sets and Systems, 157 (2006), 5273. ##[23] A. K. Majumder and A. K. Bhunia,Genetic algorithm for asymatric traveling salesman problem with imprecise travel times, Journal of Computational and Applied Mathematics,235(9)(2011), 30633078. ##[24] Z. Michalewicz,Genetic Algorithms + data structures= evolution programs, Springer, Berlin,1992. ##[25] L. A. MoncayoMartinez and D. Z. Zhang,Multiobjective ant colony optimisation : a metaheuristic approach to supply chain design, International Journal of Production Economics,1(131)(2011), 407420. ##[26] C. Moon, J. Ki, G. Choi and Y. Seo,An ecient genetic algorithm for the traveling salesman problem with precedence constraints, European Journal of Operational Research, 140(2002),606617. ##[27] H. NezamabadiPour, S. Yazdani, M. M. Farsangi and M. Neyestani,A solution to an economic dispatch problem by a fuzzy adaptive genetic algorithm, Iranian Journal of Fuzzy Systems,8(3)(2011), 121. ##[28] H. D. Nguyen, I. Yoshihara, K. Yamamori and M. Yasunaga,Implementation of an effective hybrid GA for large scale traveling salesman problem, IEEE Transactions on Systems, Man,and Cybernatics,37(1) 007), 9299. ##[29] I. M. Oliver, D. J. Smith and J. R. C. Holland,A study of permutation crossover operators on the traveling salesman problem, In: Proceedings of the Second International Conference on Genetic Algorithms (ICGA'87), Massachusetts Institute of Technology, Cambridge, MA,(1987), 224230. ##[30] M. Padberg and G. Rinaldi,Optimization of a 532city symmetric traveling salesman problem by branch and cut, Operations Research Letters, 6(1) (1987), 17. ##[31] M. W. Padberg and S. Hong,On the symmetric traveling salesman problem: a computational study, Mathematical Programming Studies, 12 (80), 78107. ##[32] H. L. Petersen. and O. B. G. Madsen,The double travelling salesman problem with multiple stackformulation and heuristic solution approaches, European Journal of Operational Research,198 (2009), 339347. ##[33] A. Vescan and CM. Pintea,Ant colony componentbased system for travelling salesman problem, Applied Mathematical Sciences, 1(28) (2007), 13471357. ##[34] J. Wang, J. Huang, S. Rao, S. Xue and J. Yin,An adaptive genetic algorithm for solving traveling salesman problem, SpringerVerlag Berlin Heidelberg 2008 , ICIC 2008, LNAI 5227,(2008), 182189. ##[35] L. Yang, X. Li, Z. Gao and K. Li,A fuzzy minimum risk model for the railway transportation planning problem, Iranian Journal of Fuzzy Systems 8(4) (2011), 3960. ##[36] L. A. Zadeh,Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems,1(1978), 328. ##[37] H. J. Zimmermann,Fuzzy set theory and its applications, Allied Publishers Limited, India,1996. ##]
A COGNITIVE STYLE AND AGGREGATION OPERATOR
MODEL: A LINGUISTIC APPROACH FOR CLASSIFICATION
AND SELECTION OF THE AGGREGATION OPERATORS
A COGNITIVE STYLE AND AGGREGATION OPERATOR
MODEL: A LINGUISTIC APPROACH FOR CLASSIFICATION
AND SELECTION OF THE AGGREGATION OPERATORS
2
2
Aggregation operators (AOs) have been studied by many schol ars. As many AOs are proposed, there is still lacking approach to classify the categories of AO, and to select the appropriate AO within the AO candidates. In this research, each AO can be regarded as a cognitive style or individual dierence. A Cognitive Style and Aggregation Operator (CSAO) model is pro posed to analyze the mapping relationship between the aggregation operators and the cognitive styles represented by the decision attitudes. Four algorithms are proposed for CSAO: CSAO1, CSAO2 and two selection strategies on the basis of CSAO1 and CSAO2. The numerical examples illustrate how the choice of the aggregation operators on the basis of the decision attitudes can be determined by the selection strategies of CSAO1 and CSAO2. The CSAO model can be applied to decision making systems with the selection problems of the appropriate aggregation operators with consideration of the cognitive styles of the decision makers.
1
Aggregation operators (AOs) have been studied by many schol ars. As many AOs are proposed, there is still lacking approach to classify the categories of AO, and to select the appropriate AO within the AO candidates. In this research, each AO can be regarded as a cognitive style or individual dierence. A Cognitive Style and Aggregation Operator (CSAO) model is pro posed to analyze the mapping relationship between the aggregation operators and the cognitive styles represented by the decision attitudes. Four algorithms are proposed for CSAO: CSAO1, CSAO2 and two selection strategies on the basis of CSAO1 and CSAO2. The numerical examples illustrate how the choice of the aggregation operators on the basis of the decision attitudes can be determined by the selection strategies of CSAO1 and CSAO2. The CSAO model can be applied to decision making systems with the selection problems of the appropriate aggregation operators with consideration of the cognitive styles of the decision makers.
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60
Kevin Kam Fung
Yuen
Kevin Kam Fung
Yuen
Department of Computer science and Software Engineering,
Xi'an JiaotongLiverpool University, 111 Ren Ai Road, Suzhou Industrial Park, Suzhou,
Jiangsu Province, 215123, P. R. China
Department of Computer science and Software
China
kevinkf.yuen@gmail.com
Cognitive styles
Aggregation operators
Information fusion
Decision attitudes
Decision making
[[1] B. S. Ahn and H. Park,Leastsquared ordered weighted averaging operator weights, International Journal of Intelligent Systems,23 (2008), 3349. ##[2] G. W. Allport,Personality: a psychological interpretation, Holt & Co, New York, 1937. ##[3] G. R. Amin and A. Emrouznejad,Parametric aggregation in ordered weighted averaging,International Journal of Approximate Reasoning,52 (2011), 819827. ##[4] N. Braisby and A. Gellatly,Foundations of cognitive psychology, in Braisby, N. and Gellatly,A. , eds., Cognitive Psychology, Oxford University Press Inc., Chapter 1, (2005), 132. ##[5] P. S. Bullen, D. S. Mitrinovic and O. M. Vasic,Means and their inequalities, D. Reidel Publishing Company, Dordrecht, 1988. ##[6] N. Cagman and S. Enginoglu,Fuzzy soft matrix theory and its application in decision making Iranian Journal of Fuzzy Systems,(2012), 109119. ##[7] T. Calvo and R. Mesiar,Weighted triangular normsbased aggregation operators, Fuzzy Sets and Systems,137 (2003), 310. ##[8] M. DetynieckI,Mathematical aggregation operators and their application to video querying,Doctoral Thesis Research Report 20012002, Laboratoire dInformatique de Paris, 2000. ##[9] D. Dubois, H. Prade and C. Testemale,Weighted fuzzy patternmatching, Fuzzy Sets and Systems,28 (1988), 313331. ##[10] D. Dubois, H. Fargier and H. Prade,Re nements of the maximin approach to decisionmaking in a fuzzy environment, Fuzzy Sets and Systems, 81 (1996), 103122. ##[11] D. Dubois and H. Prade,An introduction to bipolar representations of information and preference, International Journal of Intelligent Systems, 23 (2008), 866877. ##[12] M. Espinilla, J. Liu and L. Martnez,An extended hierarchical linguistic model for decisionmaking problems, Computational Intelligence, 27 (2011), 489512. ##[13] J. Fodor and M. Roubens,Fuzzy preference modeling and multicriteria decision support,Kluwer Academic Publisher, Dordrecht, 1994. ##[14] J. L. GarcaLapresta and M. MartnezPanero,Linguisticbased voting through centered OWA operators, Fuzzy Optimization and Decision Making, 8(2009), 381393. ##[15] R. R. Ghiselli and R. Mesiar,Multiattribute aggregation operators, Fuzzy Sets and Systems,181(2011), 113. ##[16] M. Grabisch, H. T. Nguyen and E. A. Walker,amentals of uncertainty calculi with applications to fuzzy inference, Kluwer Academics Publishers, Dordrecht, 1995. ##[17] F. Herrera, S. Alonso, F. Chiclana and E. HerreraViedma,Computing with words in decision making: foundations, trends and prospects, Fuzzy Optimization and Decision Making, 8(2009), 337364. ##[18] F. Herrera and L. Martinez,A 2tuple fuzzy linguistic representation model for computing with words, IEEE Transactions on Fuzzy Systems, 8 (2000), 746752. ##[19] J. L. Marichal,Aggregation operators for multicriteria decision aid PhD. Thesis, University of Lige, Belgium, 1998. ##[20] J. Martn, G. Mayor and O. Valero,On aggregation of normed structures, Mathematical and Computer Modelling,54 (2011), 815827. ##[21] L. Martnez, D. Ruan and F. Herrera,Computing with words in decision support systems:an overview on models and applications, International Journal of Computational Intelligence Systems,3 (2010), 382395. ##[22] R. J. Riding and I. Cheema,Cognitive stylesan overview and integration, Educational Psychology,11 (1991), 193215. ##[23] R. Smolikava and M. P. Wachowiak,Aggregation operators for selection problems, Fuzzy Sets and Systems,131(2002), 2334. ##[24] Z. X. Su, G. P. Xia, M. Y. Chen and L. Wang,Induced generalized intuitionistic fuzzy OWA operator for multiattribute group decision making, Expert Systems with Applications,39(2012), 19021910. ##[25] W. Wang and X. Liu,Intuitionistic fuzzy geometric aggregation operators based on einstein operations, International Journal of Intelligent Systems, 26 (2011), 10491075. ##[26] G. Wei,Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making, Applied Soft Computing, 10 (2010), 423431. ##[27] M. Xia and Z. Xu,Entropy/cross entropybased group decision making under intuitionistic fuzzy environment, Information Fusion, 13(2012), 3147. ##[28] Z. Xu and X. Cai,Recent advances in intuitionistic fuzzy information aggregation, Fuzzy Optimization and Decision Making,9 (2010), 359381. ##[29] R. R. Yager,On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE trans. Systems, Man Cybernet., 18 (1988), 183190. ##[30] R.R. Yager,On weighted median aggregation, Internat. J. Uncertainty, Fuzziness Knowledgebased Systems,2 (1994), 101113. ##[31] R. R. Yager,On the analytic representation of Leximin ordering and its application to exible constraint propagation, European J. Oper. Res., 102 (1997), 176192. ##[32] R. R. Yager and A. Rybalov,Full reinforcement operators in aggregation techniques, IEEE Trans. On Systems, Man, and Cybernetics Part B,28 (1998), 757769. ##[33] R. R. Yager,OWA aggregation over a continuous interval argument with applications to decision making, IEEE Trans. On Systems, Man and Cybernetics Part B, 34(2004), 19521963. ##[34] R. R. Yager and A. Rybalov,Bipolar aggregation using the Uninorms, Fuzzy Optimization and Decision Making,10(2011), 5970. ##[35] K. K. F. Yuen and H. C. W. Lau,A linguisticpossibilityprobability aggregation model for decision analysis with imperfect knowledge, Applied Soft Computing, 9(2009), 575589. ##[36] K. K. F. Yuen,Selection of aggregation operators with decision attitudes, In J. Mehnen,A. Tiwari, M. Kppen and A. Saad, eds., Applications of Soft Computing: From Theory to Praxis, Advances in Intelligent and Soft Computing,58 (2009), 255264. ##[37] K. K. F. Yuen,Cognitive network process with fuzzy soft computing technique for collective decision aiding, The Hong Kong Polytechnic University, PhD. Thesis, 2009. ##[38] K. K. F. Yuen,The primitive cognitive network process: comparisons with the analytic hierarchy process, International Journal of Information Technology and Decision Making, 10(2011), 659680. ##[39] K. K. F. Yuen, Membership maximization prioritization methods for fuzzy analytic hierarchy process, Fuzzy Optimization and Decision Making, 11 (2012), 113133. ##[40] S. Zeng and W. Su,Intuitionistic fuzzy ordered weighted distance operator, KnowledgeBased Systems,24 (2011), 12241232. ##[41] H. J. Zimmermann and P. Zysno,Latent connectives in human decision making, Fuzzy Sets and ystems, 4 (1980), 3751. ##]
FUZZY GOAL PROGRAMMING TECHNIQUE TO SOLVE
MULTIOBJECTIVE TRANSPORTATION PROBLEMS WITH
SOME NONLINEAR MEMBERSHIP FUNCTIONS
FUZZY GOAL PROGRAMMING TECHNIQUE TO SOLVE
MULTIOBJECTIVE TRANSPORTATION PROBLEMS WITH
SOME NONLINEAR MEMBERSHIP FUNCTIONS
2
2
The linear multiobjective transportation problem is a special type of vector minimum problem in which constraints are all equality type and the objectives are conicting in nature. This paper presents an application of fuzzy goal programming to the linear multiobjective transportation problem. In this paper, we use a special type of nonlinear (hyperbolic and exponential) membership functions to solve multiobjective transportation problem. It gives an optimal compromise solution. The obtained result has been compared with the solution obtained by using a linear membership function. To illustrate the methodology some numerical examples are presented.
1
The linear multiobjective transportation problem is a special type of vector minimum problem in which constraints are all equality type and the objectives are conicting in nature. This paper presents an application of fuzzy goal programming to the linear multiobjective transportation problem. In this paper, we use a special type of nonlinear (hyperbolic and exponential) membership functions to solve multiobjective transportation problem. It gives an optimal compromise solution. The obtained result has been compared with the solution obtained by using a linear membership function. To illustrate the methodology some numerical examples are presented.
61
74
Maryam
Zangiabadi
Maryam
Zangiabadi
Department of Applied Mathematics, Faculty of Mathematical
Sciences, Shahrekord University, P.O. Box 115, Shahrekord, Iran
Department of Applied Mathematics, Faculty
Iran
zangiabadim@sci.sku.ac.ir
Hamid Reza
Maleki
Hamid Reza
Maleki
Department of Basic Sciences, Shiraz University of Technology,
Shiraz, Iran
Department of Basic Sciences, Shiraz University
Iran
maleki@sutech.ac.ir
Multiobjective decision making
Goal programming
Transportation problem
Membership Function
Fuzzy programming
[[1] W. F. Abd ElWahed and S. M. Lee,Interactive fuzzy goal programming for multiobjective transportation problems, Omega, 34 (2006), 158166. ##[2] R. S. Aenaida and N. W. Kwak,A linear goal programming for transshipment problems with exible supply and demand constraints, Fuzzy Sets and Systems, 45 (1994), 215224. ##[3] A. K. Bit, M. P. Biswal and S. S. Alam,Fuzzy programming approach to multicriteria decision making transportation problem, Fuzzy Sets and Systems, 50 (1992), 135141. ##[4] J. Brito, J. A. Moreno and J. L. Verdegay,Transport route planning models based on fuzzy approach, Iranian Journal of Fuzzy Systems, 9(1) (2012), 141{158. ##[5] S. Chanas and D. Kuchta,A concept of the optimal solution of the transportation problem with fuzzy cost coecients, Fuzzy Sets and Systems, 28(1996), 299305. ##[6] A. Charnes and W. W. Cooper,Management models and industrial applications of linear programming, Wiley, New York, 1961. ##[7] A. Charnes, W. W. Cooper and A. Henderson,An introduction to linear programming, Wiley,New York, 1953. ##[8] J. Current and H. Min,Multiobjective design of transportation networks: taxonomy and annotation, European J. Oper. Res., 26 (1986), 187201. ##[9] G. B. Dantzig,Linear programming and extensions, Princeton University Press, Princeton,N J, 1963. ##[10] A. K. Dhingra and H. Moskowitz,Application of fuzzy theories to multiple objective decision making in system design, European J. Oper. Res., 55 (1991), 348361. ##[11] J. A. Diaz,Solving multiobjective transportation problem, Ekonom.Mat. Obzor., 14 (1978),267274. ##[12] J. A. Diaz,Finding a complete description of all ecient solutions to a multiobjective transportation problem, Ekonom.Mat. Obzor., 15(1979), 6273. ##[13] W. Edwards,How to use multiattribute utility measurement for social decision making, IEEE Trans. Systems Man Cybernet.,7 (1977), 326340. ##[14] A. Gupta and G. W. Evans,A goal programming model for the operation of closedloop supply chains, Engineering Optimization, 41 2009), 713735. ##[15] E. L. Hannan,On fuzzy goal programming, Decision Sci., 12 (1981), 522531. ##[16] H. Hassanpour, H. R. Maleki and M. A. Yaghoobi,uzzy linear regression model with crisp coecients: a Goal programming appproach, Iranian Journal of Fuzzy Systems, 7(2) (2010),1939. ##[17] ] F. L. Hitchcock,The distribution of a product from several sources to numerous localities,J. Math. Phys.,20 (1941), 224230. ##[18] H. Isermann,The enumeration of all ecient solutions for a linear multiobjective trans portation problem, Naval Res. Logist. Quart., 2 (1979), 123139. ##[19] F. Jimenez and J. L. Verdegay,Solving fuzzy solid transportation problems by an evolutionary algorithm based parametric approach, European Journal of Operational Research, 117 (1999),485510. ##[20] Amarpreet Kaur and Amit Kumar,A new method for solving fuzzy transportation problems using ranking function, Applied Mathematical Modelling, 35 (2011), 56525661. ##[21] ] H. Leberling,On nding compromise solutions for multicriteria problems using the fuzzy minoperator, Fuzzy Sets and Systems, 6 (1981), 105118. ##[22] S. M. Lee and L. J. Moore,Optimizing transportation problems with multiple objectives,AIEE Transactions,5 (1973), 333338. ##[23] L. S. Li and K. K. Lai,A fuzzy approach to the multiobjective transportation problem, Computers and Operations Research,27 2000), 4357. ##[24] R. H. Mohamed,The relationship between goal programming and fuzzy programming, Fuzzy Sets and Systems,89 (1997), 215222. ##[25] B. B. Pal, B. N. Moitra and U. Maulik,A goal programming procedure for fuzzy multiobjectivelinear programming problem, Fuzzy Sets and Systems, 139(2003), 395405. ##[26] D. Peidro and P. Vasant,Transportation planning with modi ed scurve membership functions using an interactive fuzzy multiobjective approach, Applied Soft Computing, 11 2011),26562663. ##[27] J. L. Ringuest and D. B. Rinks,Interactive solutions for the linear multiobjective transportation problem, European J. Oper. Res., 32(1987), 96{106. ##[28] M. Sakawa,Fuzzy sets and interactive multiobjective optimization, Plenum Press, New York,1993. ##[29] R. N. Tiwari, S. Dharmar and J. R. Rao,Fuzzy goal programmingan additive model, Fuzzy Sets and Systems,24(1987), 2734. ##[30] R. Verma, M. P. Biswal and A. Biswas,Fuzzy programming technique to solve multi bjectivetransportation problem with some nonlinear membership functions , Fuzzy Sets and Systems, 91(1997), 37 43. ##[31] M. A. Yaghoobi and M. Tamiz,A short note on the relationship between goal programming and fuzzy programming for vectormaximum problems, Iranian Journal of Fuzzy Systems,2(2)(1979), 3136. ##[32] M. Zangiabadi and H. R. Maleki,Fuzzy goal programming for multiobjective transportation problems, J. Appl. Math. and Computing, 24(12) (2007), 449460. ##[33] H. J. Zimmermann,Application of fuzzy set theory to mathematical programming, Information Sciences, 36 (1985), 2958. ##]
MINIMIZATION OF DETERMINISTIC FINITE AUTOMATA
WITH VAGUE (FINAL) STATES AND INTUITIONISTIC
FUZZY (FINAL) STATES
MINIMIZATION OF DETERMINISTIC FINITE AUTOMATA
WITH VAGUE (FINAL) STATES AND INTUITIONISTIC
FUZZY (FINAL) STATES
2
2
In this paper, relations among the membership values of gener alized fuzzy languages such as intuitionistic fuzzy language, intervalvalued fuzzy language and vague language are studied. It will aid in studying the properties of one language when the properties of another are known. Further, existence of a minimized nite automaton with vague ( final) states for any vague regular language recognized by a nite automaton with vague ( final) states is shown in this paper. Finally, an ecient algorithm is given for minimizing the nite automaton with vague ( final) states. Similarly, it can be shown for intuitionistic fuzzy regular language. These may contribute to a better understanding of the role of nite automaton with vague ( final) states or the nite automaton with intuitionistic fuzzy ( final) states while studying lexical analysis, decision making etc.
1
In this paper, relations among the membership values of gener alized fuzzy languages such as intuitionistic fuzzy language, intervalvalued fuzzy language and vague language are studied. It will aid in studying the properties of one language when the properties of another are known. Further, existence of a minimized nite automaton with vague (final) states for any vague regular language recognized by a nite automaton with vague ( nal) states is shown in this paper. Finally, an ecient algorithm is given for minimizing the nite automaton with vague ( final) states. Similarly, it can be shown for intuitionistic fuzzy regular language. These may contribute to a better understanding of the role of nite automaton with vague ( final) states or the nite automaton with intuitionistic fuzzy ( final) states while studying lexical analysis, decision making etc.
75
88
Alka
Choubey
Alka
Choubey
Mathematics Department, Jaypee Institute of Information Technol
ogy, A10, Sector62, Noida201307 (U. P.), India
Mathematics Department, Jaypee Institute
India
alka.choubey@jiit.ac.in, alka.choubey@gmail.com
K. M.
Ravi
K. M.
Ravi
Department of Mathematics, JSS Academy of Technical Education, C
20/1, Sector62, Noida201301 (U. P), India
Department of Mathematics, JSS Academy of
India
rv.km19@gmail.com, rv km@yahoo.com
Intuitionistic fuzzy regular language
Intervalvalued fuzzy regular language
Vague regular language
Finite automaton with vague (final) states
Finite automaton with intuitionistic fuzzy (nal) states
MyhillNerode theorem
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On the Diagram of One Type Modal Operators on Intuitionistic fuzzy
sets: Last expanding with $Z_{alpha ,beta }^{omega ,theta
On the Diagram of One Type Modal Operators on Intuitionistic fuzzy
sets: Last expanding with $Z_{alpha ,beta }^{omega ,theta
2
2
Intuitionistic Fuzzy Modal Operator was defined by Atanassov in cite{at3}in 1999. In 2001, cite{at4}, he introduced the generalization of thesemodal operators. After this study, in 2004, Dencheva cite{dencheva} definedsecond extension of these operators. In 2006, the third extension of thesewas defined in cite{at6} by Atanassov. In 2007,cite{gc1}, the authorintroduced a new operator over Intuitionistic Fuzzy Sets which is ageneralization of Atanassov's and Dencheva's operators. At the same year,Atanassov defined an operator which is an extension of all the operatorsdefined until 2007. The diagram of One Type Modal Operators onIntuitionistic Fuzzy Sets was introduced first in 2007 by Atanassovcite{at10}. In 2008, Atanassov defined the most general operator and in2010 the author expanded the diagram of One Type Modal Operators onIntuitionistic Fuzzy Sets with the operator $Z_{alpha ,beta }^{omega }$.Some relationships among these operators were studied by several researchers%cite{at5}cite{at8} cite{gc1}, cite{gc3}, cite{dencheva} cite%{narayanan}.The aim of this paper is to expand the diagram of one type modal operatorsover intuitionistic fuzzy sets . For this purpose, we defined a new modaloparator $Z_{alpha ,beta }^{omega ,theta }$ over intuitionistic fuzzysets. It is shown that this oparator is the generalization of the operators$Z_{alpha ,beta }^{omega },E_{alpha ,beta },boxplus _{alpha ,beta},boxtimes _{alpha ,beta }.$
1
Intuitionistic Fuzzy Modal Operator was defined by Atanassov in cite{at3}in 1999. In 2001, cite{at4}, he introduced the generalization of thesemodal operators. After this study, in 2004, Dencheva cite{dencheva} definedsecond extension of these operators. In 2006, the third extension of thesewas defined in cite{at6} by Atanassov. In 2007,cite{gc1}, the authorintroduced a new operator over Intuitionistic Fuzzy Sets which is ageneralization of Atanassov's and Dencheva's operators. At the same year,Atanassov defined an operator which is an extension of all the operatorsdefined until 2007. The diagram of One Type Modal Operators onIntuitionistic Fuzzy Sets was introduced first in 2007 by Atanassovcite{at10}. In 2008, Atanassov defined the most general operator and in2010 the author expanded the diagram of One Type Modal Operators onIntuitionistic Fuzzy Sets with the operator $Z_{alpha ,beta }^{omega }$.Some relationships among these operators were studied by several researchers%cite{at5}cite{at8} cite{gc1}, cite{gc3}, cite{dencheva} cite%{narayanan}.The aim of this paper is to expand the diagram of one type modal operatorsover intuitionistic fuzzy sets . For this purpose, we defined a new modaloparator $Z_{alpha ,beta }^{omega ,theta }$ over intuitionistic fuzzysets. It is shown that this oparator is the generalization of the operators$Z_{alpha ,beta }^{omega },E_{alpha ,beta },boxplus _{alpha ,beta},boxtimes _{alpha ,beta }.$
89
106
g.
cuvalcioglu
g.
cuvalcioglu
department of mathematics, university of mersin, ciftlikkoy, 33016,
mersin turkey
department of mathematics, university of
Turkey
gcuvalcioglu@mersin.edu.tr
Modal operator
$Z_{alpha
beta }^{omega
theta }$ operator
Modal operator diagram
[bibitem{at1} K. T. Atanassov, emph{Intuitionistic fuzzy sets}, VII ITKR's Session,##Sofia, June 1983. ##bibitem{at2} K. T. Atanassov, emph{Intuitionistic fuzzy sets}, Fuzzy Sets and##Systems, textbf{20} (1986), 8796. ##bibitem{at3} K. T. Atanassov, emph{Intuitionistic fuzzy sets}, PhiysicaVerlag,##Heidelberg, NewYork, 1999. ##bibitem{at4} K. T. Atanassov, emph{Remark on two operations over intuitionistic##fuzzy sets,} Int. J. of Unceratanity, Fuzzyness and Knowledge Syst.,##textbf{9(1)} (2001), 7175. ##bibitem{at5} K. T. Atanassov, emph{On the type of intuitionistic fuzzy modal##operators}, NIFS, textbf{11(5)} (2005), 2428. ##bibitem{at6} K. T. Atanassov, emph{The most general form of one type of##intuitionistic fuzzy modal operators}, NIFS, textbf{12(2)} (2006), 3638. ##bibitem{at7} K. T. Atanassov, emph{Some properties of the operators from one type##of intuitionistic fuzzy modal operators}, Advanced Studies on Contemporary##Mathematics, textbf{15(1)} (2007), 1320. ##bibitem{at8} K. T. Atanassov, emph{The most general form of one type of##intuitionistic fuzzy modal operators, part 2}, NIFS, textbf{14(1)} (2008), 2732. ##bibitem{at9} K.T. Atanassov, emph{Theorem for equivalence of the two most##general intuitionistic fuzzy modal operators}, NIFS, textbf{15(1)}(2008), 2631. ##bibitem{at10} K. T. Atanassov, emph{25 years of intuitionistic fuzzy sets, or:##the most important results and mistakes of mine}, 7 th Int. workshop on IFSs##and gen. nets. , Poland, 2008. ##bibitem{gc1} G. c{C}uvalci ou{g}lu, emph{Some properties of $E_{alpha##,beta }$ operator}, Advanced Studies on Contemporary Mathematics, textbf{14(2)} (2007), 305310. ##bibitem{gc2} G. c{C}uvalci ou{g}lu, emph{Expand the modal operator diagram##with $Z_{alpha ,beta }^{omega },$}, Proc. Jangjeon Math. Soc., textbf{13(3)} (2010), 403412 ##bibitem{gc3} G. c{C}uvalci ou{g}lu, S. Yi lmaz, emph{Some properties of ##OTMOs on IFSs, Advanced Studies on Contemporary Mathematics}, textbf{14(2)} (2010),##bibitem{dencheva} K. Dencheva, emph{Extension of intuitionistic fuzzy modal##operators $boxplus $ and $boxtimes ,$}, Proc.of the Second Int. IEEE Symp.##Intelligent systems, Varna, June 2224, textbf{3} (2004), 2122. ##bibitem{doycheva} B. Doycheva, emph{Inequalities with intuitionistic fuzzy##topological and G"{o}khan c{C}uvalci ou{g}lu's operators}, NIFS, textbf{14(1)} (2008), 2022. ##bibitem{hasan} A. Hasankhani, A. Nazari and M. Saheli, emph{Some properties of##fuzzy Hilbert spaces and norm of operators}, Iranian Journal of Fuzzy Systems,##textbf{7(3)} (2010), 129157. ##bibitem{li} D. Li, F. Shan and C. Cheng, emph{On properties of four IFS##operators}, Fuzzy Sets and Systems, textbf{154} (2005), 151155. ##bibitem{luo} X. Luo and J. Fang, emph{Fuzzifying closure systems and closure##operators}, Iranian Journal of Fuzzy Systems, textbf{8(1)} (2011), 7794. ##bibitem{narayanan} A. Narayanan, S. Vijayabalaji and N. Thillaigovindan, emph{Intuitionistic fuzzy bounded linear operators}, Iranian Journal of Fuzzy Systems, textbf{4(1)} (2007), 89101. ##bibitem{zadeh} L. A. Zadeh, emph{Fuzzy sets}, Information and Control, textbf{8} (1965) ,##]
FUZZY INTEGRODIFFERENTIAL EQUATIONS: DISCRETE
SOLUTION AND ERROR ESTIMATION
FUZZY INTEGRODIFFERENTIAL EQUATIONS: DISCRETE
SOLUTION AND ERROR ESTIMATION
2
2
This paper investigates existence and uniqueness results for the first order fuzzy integrodifferential equations. Then numerical results and error bound based on the left rectangular quadrature rule, trapezoidal rule and a hybrid of them are obtained. Finally an example is given to illustrate the performance of the methods.
1
This paper investigates existence and uniqueness results for the first order fuzzy integrodifferential equations. Then numerical results and error bound based on the left rectangular quadrature rule, trapezoidal rule and a hybrid of them are obtained. Finally an example is given to illustrate the performance of the methods.
107
122
Masoumeh
Zeinali
Masoumeh
Zeinali
Faculty of mathematical sciences, University of Tabriz, Tabriz,
Iran
Faculty of mathematical sciences, University
Iran
zeynali@tabrizu.ac.ir
Sedaghat
Shahmorad
Sedaghat
Shahmorad
Faculty of mathematical sciences, University of Tabriz, Tabriz,
Iran
Faculty of mathematical sciences, University
Iran
shahmorad@tabrizu.ac.ir
Kamal
Mirnia
Kamal
Mirnia
Faculty of mathematical sciences, University of Tabriz, Tabriz, Iran
Faculty of mathematical sciences, University
Iran
mirniakam@tabrizu.ac.ir
Fuzzy integrodifferential equation
Discrete solution
Fuzzy quadrature rule
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SETNORM EXHAUSTIVE SET MULTIFUNCTIONS
SETNORM EXHAUSTIVE SET MULTIFUNCTIONS
2
2
In this paper we present some properties of setnorm exhaustive set multifunctions and also of atoms and pseudoatoms of set multifunctions taking values in the family of nonempty subsets of a commutative semigroup with unity.
1
In this paper we present some properties of setnorm exhaustive set multifunctions and also of atoms and pseudoatoms of set multifunctions taking values in the family of nonempty subsets of a commutative semigroup with unity.
123
134
Anca
Croitoru
Anca
Croitoru
Faculty of Mathematics, "A.I. Cuza" University, Bd. Carol I, no 11,
Iasi700506, Romania
Faculty of Mathematics, "A.I. Cuza" University,
Romania
croitoru@uaic.ro
Alina
Gavrilut
Alina
Gavrilut
Faculty of Mathematics, "A.I. Cuza" University, Bd. Carol I, no
11, Iasi700506, Romania
Faculty of Mathematics, "A.I. Cuza" University,
Romania
gavrilut@uaic.ro
Setnorm
Exhaustive
Continuous
Nullnulladditive
Atom
Pseudo atom
[[1] S. Asahina, K. Uchino and T. Murofushi,Relationship among continuity conditions and nulladditivity ditions in nonadditive measure theory, Fuzzy Sets and Systems, 157 (2006),691698. ##[2] R. J. Aumann and L. S. Shapley,Values of nonatomic games, Princeton University Press,Princeton, New Jersey, 1974. ##[3] I. Chitescu,Finitely purely atomic measures: coincidence and rigidity properties, Rendiconti del Circolo Matematico di Palermo, Serie II, Tomo L, (2001), 455476. ##[4] G. Choquet,Theory of capacities, Ann. Inst. Fourier (Grenoble), 5 (19531954), 131292. ##[5] A. Croitoru,Setnorm continuity of set multifunctions, ROMAI Journal, 6 (2010), 4756. ##[6] A. Croitoru, A. Gavrilut, N. E. Mastorakis and G. Gavrilut On dierent types of nonadditiveset multifunctions,WSEAS Transactions on Mathematics, 8 (2009), 246257. ##[7] A. Daneshgar and A. Hashemi,Fuzzy sets from a metasystemtheoretic point of view, Iranian Journal of Fuzzy Systems,3(2) (2006), 120. ##[8] A. P. Dempster,Upper and lower probabilities induced by a multivalued mapping, Ann. Mat.Statist.,38(1967), 325339. ##[9] D. Denneberg,Nonadditive Measure and Integral, Kluwer Academic Publishers, Dorrecht/Boston/London, 1994. ##[10] L. Drewnowski,Topological rings of sets, continuous set functions. Integration, I, II, III,Bull. Acad. Polon. Sci. Ser. Math. Astron. Phy, s20(1972), 269286. ##[11] D. Dubois and H. Prade, Fuzzy sets and systems. Theory and applications, Academic Press,New York, 1980. ##[12] T. Funiokova,LKInterior systems of "almost open" Lsets, Iranian JournaL of Fuzzy Systems,4(2)(2007), 47{55. ##[13] A. Gavrilut,Nonatomicity and the Darboux property for fuzzy and nonfuzzy Borel/Bairemultivalued set functions, Fuzzy Sets and Systems, 160 2009), 13081317. ##[14] A. Gavrilut and A. Croitoruonatomicity for fuzzy and nonfuzzy multivalued set functions,Fuzzy Sets and Systems,160(2009), 21062116. ##[15] A. Gavrilut and A. Croitoru, Pseudoatoms and Darboux property for set multifunction,Fuzzy Sets and Systems,(2010), 28972908. ##[16] J. Li,On Egoro theorem on fuzzy measure spaces, Fuzzy Sets and Systems, 135 (2003),367375. ##[17] F. Merghadi and A. Aliouche,A related xed point theorem in n fuzzy metric spaces, Iranian Journal of Fuzzy Systems,7(3) (2010), 7386. ##[18] E. Pap,Nulladditive set functions, Kluwer Academic Publishers, Dordrecht, 1995. ##[19] A. M. Precupanu,On the set valued additive and subadditive set functions, An. St. UniIa29(1984), 4148. ##[20] G. Shafer,A Mathematical theory of evidence, Princeton University Press, Princeton, N. J.,1976. ##[21] M. Sugeno,Theory of fuzzy integrals and its applications, PhD. Thesis, Tokyo Institute ofTechnology, 1974. ##[22] H. Suzuki,Atoms of fuzzy measures and fuzzy integrals, Fuzzy Sets and Systems, 41 (1991),329342. ##[23] S. M. Vaezpour and F. Karini,tBest approximation in fuzzy normed spaces, Iranian Journal of Fuzzy Systems,5(2) (2008), 9399. ##[24] G. F. Wen, F. G. Shi and H.Y. Li,Almost Scompactness in Ltopological spaces, Iranian Journal of Fuzzy Systems,5(3) (2008), 3144. ##[25] C. Wu and S. Bo,Pseudoatoms of fuzzy and nonfuzzy measures , Fuzzy Sets and Systems,1582007), 12581272. ##[26] L. A. Zadeh,Fuzzy sets, Information and Control, 8 (1965), 338{353. ##]
APPROXIMATE FIXED POINT IN FUZZY NORMED SPACES
FOR NONLINEAR MAPS
APPROXIMATE FIXED POINT IN FUZZY NORMED SPACES
FOR NONLINEAR MAPS
2
2
We de ne approximate xed point in fuzzy norm spaces and prove the existence theorems, we also consider approximate pair constructive map ping and show its relation with approximate fuzzy xed point.
1
We de ne approximate xed point in fuzzy norm spaces and prove the existence theorems, we also consider approximate pair constructive map ping and show its relation with approximate fuzzy xed point.
135
142
S. A. M.
Mohseniailhosseini
S. A. M.
Mohseniailhosseini
Faculty of Mathematics, ValieAsr University of Raf
senjan, Rafsenjan, Iran
Faculty of Mathematics, ValieAsr University
Iran
amah@vru.ac.ir
H.
Mazaheri
H.
Mazaheri
Faculty of Mathematics, Yazd University, Yazd, Iran
Faculty of Mathematics, Yazd University,
Iran
hmazaheri@yazduni.ac.ir
M. A.
Dehghan
M. A.
Dehghan
Faculty of Mathematics, ValieAsr University of Rafsenjan, Raf
senjan, Iran
Faculty of Mathematics, ValieAsr University
Iran
dehghan@vru.ac.ir
Fuzzy norm space
$F^z$approximate fixed point
Diameter $F^z$approximate fixed point
[[1] I. Altun,Some xed point theorems for single and multivalued mappings on ordered nonarchimeden fuzzy metric spaces, Iranian Journal of Fuzzy Systems, 7(1) (2008), 4962. ##[2] T. Bag and S. K. Samanta,Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math.,11(3) (2003), 687705. ##[3] T. Bag and S. K. Samanta,Fuzzy bounded linear operators, Fuzzy Sets and systems, 151(3)(2005), 513547. ##[4] T. Bag and S. K. Samanta,Some xed point theorems in fuzzy normed linear spaces, Information ciences, 177 (2007), 32713289. ##[5] F. E. Browder,Nonexpansive nonlinear operators in a Banach spaces, Proc. Natl. Acad. Sci.USA,54 (1965), 10411044. ##[6] M. Cancan,Browders xed point theorem and some interesting results in intuitionistic fuzzy normed spaces, Fixed Point Theory and Applications, Article ID 642303, 11 pages doi:10.1155/(2010)/642303, 2010. ##[7] L. Cadariu and V. Radu,On the stability of the Cauchy functional equation: a xed point approach, in Iteration Theory, Grazer Math. Ber., KarlFranzensUniversitaet, Graz, Austria,346(2004), 4352. ##[8] A. Chitra and P. V. Mordeson,Fuzzy linear operators and fuzzy normed linear spaces, Bull.Cal. Math. Soc.,74 (1969), 660665. ##[9] R. Espinola,A new approach to relatively nonexpansive mappings, Proc. Amer. Math. Soc.,136(6) (2008), 19871995. ##[10] I. Golet,On fuzzy normed spaces, Southest Asia Bull. Math., 31(2) (2007), 245254. ##[11] M. Grabic,Fixed points in fuzzy metric spaces, Fuzzy Sets ans Systems, 27(3) (1988), 385389. ##[12] M. Marudai and P. Vijayaraju,Fixed point theorems for fuzzy mapping, Fuzzy Sets and Systems,135(3) (2003), 402408. ##[13] F. Merghadi and A. Aliouche,A related xed point theorem in n fuzzy metric spaces, Iranian Journal of Fuzzy Systems,7(3) (2010), 7386. ##[14] M. Ra and M. S. M. Noorani,Fixed point theorem on intuitionistic fuzzy metric space,Iranian Journal of Fuzzy Systems,3(1) (2006), 2329. ##[15] R. Saadati, S. M. Vaezpour and Y. J. Cho,Quicksort algorithm: application of a fixedpoint theorem in intuitionistic fuzzy quasimetric spaces at a domain of words, Journal of Computational and Applied Mathematics,228(1) (2009), 219225. ##[16] Krishnapal Singh Sisodia, M. S. Rathore, Deepak Singh and Surendra Singh Khichi,A common xed point theorem in fuzzy metric spaces, Int. Journal of Math. Analysis, 5(17) (2011),819826. ##[17] T. Zikic,On xed point theorems of Gregori and Sapena, Fuzzy Sets and Systems, 144(3)(2004), 421429.##]
WEAK AND STRONG DUALITY THEOREMS FOR FUZZY
CONIC OPTIMIZATION PROBLEMS
WEAK AND STRONG DUALITY THEOREMS FOR FUZZY
CONIC OPTIMIZATION PROBLEMS
2
2
The objective of this paper is to deal with the fuzzy conic program ming problems. The aim here is to derive weak and strong duality theorems for a general fuzzy conic programming. Toward this end, The convexitylike concept of fuzzy mappings is introduced and then a speci c ordering cone is established based on the parameterized representation of fuzzy numbers. Un der this setting, duality theorems are extended from crisp conic optimization problems to fuzzy ones.
1
The objective of this paper is to deal with the fuzzy conic program ming problems. The aim here is to derive weak and strong duality theorems for a general fuzzy conic programming. Toward this end, The convexitylike concept of fuzzy mappings is introduced and then a speci c ordering cone is established based on the parameterized representation of fuzzy numbers. Un der this setting, duality theorems are extended from crisp conic optimization problems to fuzzy ones.
143
152
B.
Farhadinia
B.
Farhadinia
Department of Mathematics, Quchan Institute of Engineering and
Technology, Iran,
Department of Mathematics, Quchan Institute
Iran
bfarhadinia@yahoo.com.au
A. V.
Kamyad
A. V.
Kamyad
Department of Mathematics, Ferdowsi University of Mashhad, Iran,
Department of Mathematics, Ferdowsi University
Iran
kamyad@math.um.ac.ir
Fuzzy conic optimization problem
Fuzzy number
Weak and strong duality theorems
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