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New Approach to Exponential Stability Analysis and Stabilization for Delayed TS Fuzzy Markovian Jump Systems
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This paper is concerned with delaydependent exponential stability analysis and stabilization for continuoustime TS fuzzy Markovian jump systems with modedependent timevarying delay. By constructing a novel LyapunovKrasovskii functional and utilizing some advanced techniques, less conservative conditions are presented to guarantee the closedloop system is meansquare exponentially stable. Then, the stabilization conditions are derived and the fuzzy controller can be obtained by solving a set solutions of LMIs. The upper bound of timedelay that the system can be stabilized is given by using an optimal algorithm. Two examples are presented to illustrate the effectiveness and potential of our methods.
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19


Liu
Cui
School of Computer Science and Information Engineering, Shanghai Institute of Technology, No.100 Haiquan Road, Fengxian Distinct, 201418, Shanghai ,
China
School of Computer Science and Information
China
cuiliu8475@msn.com


Yanchai
Liu
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, No.800 Dongchuan Road, Minhang Distinct, 200240, Shanghai, China
School of Aeronautics and Astronautics, Shanghai
China


Yueying
Wang
School of Aeronautics and Astronautics, Shanghai Jiao Tong Uni
versity, No.800 Dongchuan Road, Minhang Distinct, 200240, Shanghai, China
School of Aeronautics and Astronautics, Shanghai
China
68942275@163.com


Dengping
Duan
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, No.800 Dongchuan Road, Minhang Distinct, 200240, Shanghai, China
School of Aeronautics and Astronautics, Shanghai
China
ddp@sjtu.edu.cn
Delaydependent exponential stability and stabilization
Fuzzy systems
LyapunovKrasovskii functional
Markovian jump systems
Modedependent time delay
[[1] P. Balasubramaniam and R. Sathy, Robust asymptotic stability of fuzzy Markovian jump##ing genetic regulatory networks with timevarying delays by delay decomposition approach,##Comm. Nonlinear Science Numerical Simul., 16(2) (2011), 928–939.##[2] P. Balasubramaniam and T. Senthilkumar, Delayrangedependent robust stabilization and##H∞ control for nonlinear uncertain stochastic fuzzy systems with modedependent time delays##and Markovian jump parameters, Int. J. Syst. Sci.,(2012), 1–14.##[3] L. J. Banu and P. Balasubramaniam, Admissibility analysis for discretetime singular systems##with timevarying delays by adopting the statespace TakagiSugeno fuzzy model, Int. J. Fuzzy##Syst., 12(3) (2015), 1–16.##[4] L. J. Banu and P. Balasubramaniam, Robust stability and stabilization analysis for discrete##time randomly switched fuzzy systems with known sojourn probabilities, Nonlinear Anal.##:Hybrid Syst., 17 (2015), 128–143.##[5] E. K. Boukas, Z. K. Liu and G. X. Liu, Delaydependent robust stability and H∞ control of##jump linear systems with timedelay, Int. J. Control, 74(4) (2001), 329–340.##[6] J. X. Dong and G. Yang, Fuzzy controller design for Markovian jump nonlinear systems, Int.##J. Control, Autom., Syst., 5(6) (2007), 712–717.##[7] Z. Y. Fei, H. J. Gao and P. Shi, New results on stabilization of Markovian jump systems with##time delays, Automatica, 45(10) (2010), 2300–2306.##[8] H. J. Gao, Z. Y. Fei, J. Lam and B. Du, Further results on exponential estimates of Markov##ian jump systems with modedepdendent timevarying delays, IEEE Trans. Autom. Control,##56(1) (2011), 223–229.##[9] K. Q. Gu, A further renement of discretized Lyapunov functional method for the stability##of timedelay systems, International Journal of Control, 74(1) (2001), 737–744.##[10] D. W. C. Ho and Y. G. Niu, Robust fuzzy design for nonlinear uncertain stochastic systems##via slidingmode control, IEEE Trans. Fuzzy Syst., 15(3) (2007), 350–358.##[11] M. Liu, P. Shi, L. X. Zhang and X. D. Zhao, Faulttolerant control for nonlinear Markovian##jump systems via proportional and derivative sliding mode observer technique, IEEE Trans.##Circuits. Syst. I. Regul., 58(5) (2011), 1109–1118.##[12] S. Nguang, W. Assawinchaichote, P. Shi and Y. Shi, Robust H∞ control design for uncertain##fuzzy systems with Markovian jumps: an LMI approach, In: American Control Conference,##(2005), 1805–1810.##[13] S. K. Nguang, P. Shi and S. Ding, Fault detection for uncertain fuzzy systems: an LMI##approach, IEEE Trans. Fuzzy Syst., 15(6) (2007), 1251–1262.##[14] P. Shi, Y. Q. Xia, G. P. Liu and D. Rees, On designing of slidingmode control for stochastic##jump systems, IEEE Trans. Autom. Control, 51(1) (2006), 97–103.##[15] Z. Shu, J. Lam and S. Y. Xu, Robust stabilization of Markovian jump delay systems with##delaydependent exponential estimates, Automatica, 42(11) (2006), 2001–2008.##[16] E. G. Tian and C. Peng, Delaydependent stability analysis and synthesis of uncertain TS##fuzzy systems with timevarying delay, Fuzzy Sets, Syst., 157 (2006), 544–559.##[17] S. C. Tong, W. Wang and L. J. Qu, Decentralized robust control for uncertain TS fuzzy##largescale systems with timedela, Int. J.Innovative Comput., 3(3) (2007), 657–672.##[18] Z. D. Wang, Y. R. Liu and X. H. Liu, Exponential stabilization of a class of stochastic sys##tem with Markovian jump parameters and modedependent mixed timedelays, IEEE Trans.##Autom. Control, 55(7) (2010), 1656–1662.##[19] H. N. Wu and K. Y. Cai, Modeindependent robust stabilization for uncertain Markovian##jump nonlinear systems via fuzzy control, IEEE Trans. Man, Cybern., 36(3) (2006), 509–##[20] L. G. Wu, P. Shi and H. J. Gao, State estimation and sliding mode control of Markovian##jump singular systems, IEEE Trans. Autom. Control, 55(5) (2010), 1213–1219.##[21] L. Q. Wu, P. Shi, H. J. Gao and C. H. Wang, Robust H∞ ltering for 2D Markovian jump##systems, Automatica, 44(7) (2008), 1849–1858. ##[22] L. G.Wu, X. J. Su, P. Shi and J. B. Qiu, A new approach to stability analysis and stabilization##of discretetime TS timevarying delay systems, IEEE Trans. Syst. Man, Cybern., 40(1)##(2011), 273–286.##[23] Z. J. Wu, X. J. Xie, P. Shi and Y. Q. Xia, Backstepping controller design for a class of##stochastic nonlinear systems with Markovian switching, Automatica, 45(4) (2009), 997–##[24] Z. J. Wu, X. J. Xie and P. Shi, Adaptive tracking for stochastic nonlinear systems with##Markovian switching, IEEE Trans. Autom. Control, 55(9) (2010), 2135–2141.##[25] Y. Q. Xia, M. Y. Fu, P. Shi, Z. J. Wu and J. H. Zhang, Adaptive backstepping controller##design for stochastic jump systems, IEEE Trans. Autom. Control, 54(12) (2009), 2853–2859.##[26] J. L. Xiong, J. Lam, H. J. Gao and D. W. C. Ho, On robust stabilization of Markovian jump##systems with uncertain switching probabilities, Automatica, 41(5) (2005), 897–903.##[27] S. Y. Xu and J. Lam, Robust H∞ control for uncertain discretetimedelay fuzzy systems via##output feedback controllers, IEEE Trans. Fuzzy Syst., 13(1) (2005), 82–93.##[28] D. Yue, and Q. L. Han, Delaydependent exponential stability of stochastic systems with##timevarying delay, nonlinearity, and Markovian switching, IEEE Trans. Autom. Control,##50(2) (2005), 217–222.##[29] Y. S. Zhang, S. Y. Xu and J. H. Zhang, Delaydependent robust H∞ control for uncertain##fuzzy Markovian jump systems, Int. J. Control, Autom., Syst., 7(4) (2009), 520–529.##[30] Y. H. Zhang, S. Y. Xu and B. Y. Zhang, Robust output feedback stabilization for uncertain##discretetime fuzzy Markovian jump systems with timevarying delays, IEEE Trans. Fuzzy##Syst., 17(2) (2009), 411–420.##[31] Y. Zhao, H. J. Gao, J. Lam and B. Z. Du, Stability and stabilization of delayed TS fuzzy##systems: a delay partitioning approach, IEEE Trans. Fuzzy Syst., 17(4) (2009), 750–762.##]
Assessing process performance with incapability index based on fuzzy critical value
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Process capability indices are considered as an important concept in statistical quality control. They have been widely used in the manufacturing industry to provide numerical measures on process performance. Moreover, some incapability indices have been introduced to account the process performance. In this paper, we focus on the one proposed by Chen ~cite{Che:Stat}. In today's modern world, accurate and flexible information is needed. So, we apply fuzzy logic to measure the process incapability. Buckley's approach is used to fuzzify this index and to make a decision on process incapability, we utilize fuzzy critical value. Numerical examples are presented to demonstrate the performance and effectiveness of the proposed index.
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34


Zainab
Abbasi Ganji
Department of Statistics, Faculty of Mathematical Sciences,
Ferdowsi University of Mashhad, Mashhad, Iran
Department of Statistics, Faculty of Mathematical
Iran
abbasiganji@stu.um.ac.ir


Bahram
Sadeghpour Gildeh
Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran
Department of Statistics, Faculty of Mathematical
Iran
sa deg hpour@umz.ac.ir
Process incapability index
Fuzzy sets
Hypothesis testing
Critical value
[[1] R. A. Boyles, The Taguchi capability index, Journal of Quality Technology, 23 (1991), 17–26.##[2] J. J. Buckley, Fuzzy statistics: Hypothesis testing, Soft Computing, 9 (2005a), 512–518.##[3] J. J. Buckley, Fuzzy statistics: Hypothesis testing, Soft Computing, 9 (2005b), 757–760.##[4] L. K. Chan, S. W. Cheng and F. A. Spiring, A new measure of process capability Cpm,##Quality Technology, 20 (1988), 162–175.##[5] K. S. Chen, Incapability index with asymmetric tolerances, Statistica Sinica, 8 (1998), 253–##[6] M. Greenwich and B. L. JahrSchaffrath, A process incapability index, International Journal##of Quality Reliability Management, 12 (1995), 58–71.##[7] T. C. Hsiang and G. Taguchi, Tutorial on quality control and assurance the Taguchi methods,##Joint Meeting of the American Statistical Association, Las Vegas, Nevada, 188 (1985).##[8] B. M. Hsu and M. H. Shu, Fuzzy inference to assess manufacturing process capability with##imprecise data, European Journal of Operational Research, (2008), 652–670.##[9] J. M. Juran, Juran;s Quality Control Handbook, 3rd edition, McGrawHill, New York, 1974.##[10] C. Kahraman and I. Kaya, Fuzzy process capability indices for quality control of irrigation##water, Stochastic Environmental Research and Risk Assessment, 23(4) (2009a), 451–462.##[11] C. Kahraman and I. Kaya, Fuzzy process accuracy index to evaluate risk assessment of##drought effects in Turkey, Human and Ecological Risk Assessment: An International Journal,##15(4) (2009b), 789–810.##[12] V. E. Kane, Process capability indices, Quality Technology, 18 (1986), 41–52.##[13] I. Kaya, The process incapability index under fuzziness with an application for decision making,##International Journal of Computational Intelligence Systems, 7(1) (2014), 114–128.##[14] I. Kaya and H. Baracli, Fuzzy process incapability index with asymmetric tolerances, Multiple##Valued Logic and Soft Computing, 18(56) (2012), 493–511.##[15] I. Kaya and C. Kahraman, Fuzzy process capability analyses: An application to teaching##processes, Journal of Intelligent and Fuzzy Systems, 19(45) (2008), 259–272.##[16] I. Kaya and C. Kahraman, Fuzzy robust process capability indices for risk assessment of air##pollution, Stochastic Environmental Research and Risk Assessment, 23(4) (2009a), 529–541.##[17] I. Kaya and C. Kahraman, Air pollution control using fuzzy process capability indices in sixsigma##approach, Human and Ecological Risk Assessment: An International Journal, 15(4)##(2009b), 689–713.##[18] I. Kaya and C. Kahraman, Development of fuzzy process accuracy index for decision making##problems, Information Sciences, 180(6) (2010a), 861–872.##[19] I. Kaya and C. Kahraman, A new perspective on fuzzy process capability indices: Robustness,##Expert Systems with Applications, 37(6) (2010b), 4593–4600.##[20] I. Kaya and C. Kahraman, Fuzzy process capability analyses with fuzzy normal distribution,##Expert Systems with Applications, 37(7) (2010c), 5390–5403. ##[21] I. Kaya and C. Kahraman, Process capability analyses with fuzzy parameters, Expert Systems##with Applications, 38(9) (2011a), 11918–11927.##[22] I. Kaya and C. Kahraman, Process capability analyses based on fuzzy measurements and##fuzzy control charts, Expert Systems with Applications, 38(4) (2011b), 3172–3184.##[23] I. Kaya and C. Kahraman, Fuzzy process capability indices with asymmetric tolerances, Expert##Systems with Applications, 38 (2011c), 14882–14890.##[24] Y. H. Lee, C. C. Wei and C. L. Chang, Fuzzy design of process tolerances to maximize process##capability, International Journal of Advanced Manufacturing Technology, 15 (1999), 655–659.##[25] H. T. Lee, Cpk index estimation using fuzzy numbers, European Journal of Operational##Research, 129 (2001), 683–688.##[26] A. Parchami, M. Mashinchi, A. R. Yavari and H. R. Maleki, Process capability indices as##fuzzy numbers, Austrian Journal of Statisics, 34(4) (2005), 391–402.##[27] A. Parchami, M. Mashinchi and H. R. Maleki, Fuzzy confidence interval for fuzzy process##capability index, Journal of Intelligent and Fuzzy Systems, 17 (2006), 287–295.##[28] A. Parchami and M. Mashinchi, Fuzzy estimation for process capability indices, Information##Sciences, 177 (2007), 1452–1462.##[29] A. Parchami and M. Mashinchi, A new generation of process capability indices, Journal of##Applied Statistics, 37(1) (2010), 77–89.##[30] Z. Ramezani, A. Parchami and M. Mashinchi, Fuzzy confidence regions for the Taguchi##capability index, International Journal of Systems Science, 42(6) (2011), 977–987.##[31] B. SadeghpourGildeh, Comparison of and process capability indices in the case of measurement##error occurrence, IFSA World Congress, Istanbul, Turkey, (2003), 563–567.##[32] B. SadeghpourGildeh and T. Angoshtari, Monitoring fuzzy capability index ˜ Cpk by using the##EWMA control chart with imprecise data, Iranian Journal of Fuzzy Systems, 10(2) (2013),##111–132.##[33] K. Vannman, A unified approach to capability indices, Statistica Sinica, (5) (1995), 805–820.##[34] C. Yongting, Fuzzy quality and analysis on fuzzy probability, Fuzzy Sets and Systems, 83##(1996), 283–290.##[35] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338–353.##[36] H. J. Zimmermann, Fuzzy set theory, John Wiley & Sons, Inc., 2 (2010), 317–332.##]
Profit maximization solid transportation problem under budget constraint using fuzzy measures
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Fixed charge solid transportation problems are formulated as profit maximization problems under a budget constraint at each destination. Here item is purchased in different depots at different prices. Accordingly the item is transported to different destinations from different depots using different vehicles. Unitsare sold from different destinations to the customers at different selling prices. Here selling prices, purchasing costs, unit transportation costs, fixed charges, sources at origins, demands at destinations, conveyances capacities are assumed to be crisp or fuzzy. Budget constraints at destinations are imposed. Itis also assumed that transported units are integer multiple of packets. So the problem is formulated as constraint optimization integer programming problem in crisp and fuzzy environments. Asoptimization of fuzzy objective as well as consideration of fuzzy constraint is not well defined, different measures possibility/necessity/credibility of fuzzy event are used to transform the problem into equivalent crisp problem. The reduced crisp problem is solved following generalized reduced gradient(GRG) method using lingo software. A dominance based genetic algorithm (DBGA) and a particle swarm optimization (PSO) technique using swap sequence are also developed for this purpose and are used to solve the model. The models are illustrated with numerical examples. The results obtained using DBGA and PSO are compared with those obtained from GRG.Moreover, a statistical analysis is presented to compare the algorithms.
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63


Pravash
Kumar Giri
Department of Applied Mathematics with Oceanology and
Computer Programming, Vidyasagar University, PaschimMedinipur, W.B. 721102, India
Department of Applied Mathematics with Oceanology
India


Manas
Kumar Maiti
Department of Mathematics, Mahishadal Raj College, Mahishadal,
PurbaMedinipur, W.B.721628, India
Department of Mathematics, Mahishadal Raj
India


Manoranjan
Maiti
Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, PaschimMedinipur, W.B. 721102, India
Department of Applied Mathematics with Oceanology
India
mmmaiti2005@yahoo.co.in
Solid transportation problem
Budget constraints
Possibility /Necessity/Credibility measure
Dominance based genetic algorithm
Particle swarm optimization
[[1] M. A. H. Akhand, S. Akter and M. A. Rashid, Velocity tentative particle swarm optimiza##tion to solve TSP, International Conference on Electrical Information and Communication##Technology (EICT), 2013.##[2] M. Bessaou and P. Siarry, A genetic algorithm with realvalue coding to optimize multimodal##continuous function, Structural Multidisciplinary Optimization, 23 (2001), 63{74. ##[3] Q. Cui and Y. Sheng, Uncertain Programming Model for Solid Transportation Problem,##Information Journal, 15 (12) (2012), 342{348.##[4] T. E. Davis and J. C. Principe,A simulated annealinglike convergence theory for the simple##genetic algorithm, In R. K. Belew, L.B. Booker (Eds.), Proceedings of the fourth international##conference on genetic algorithms , San Mateo, CA: Morgan Kaufmann, (1991), 174{181.##[5] D. Dubois and H. Prade, Fuzzy sets and systemTheory and application, Academic, New##York, 1980 .##[6] D. Dubois and H. Prade, Ranking fuzzy numbers in the setting of Possibility Theory, Information##Sciences, 30 (1983), 183{224.##[7] K. Durai Raj, A. Antony and C. Rajendran, Fast heuristic algorithms to solve a single##stage xedcharge transportation problem, International Journal of Operation Research, 6(3)##(2009), 304{329.##[8] R. C. Eberhart and J. Kennedy, A new optimizer using Particle swarm theory, In Proceedings##of the Sixth International Symphosium on micromachine and human science, (1995), 39{43.##[9] A. P. Engelbrecht,Fundamentals of Computational Swarm Intelligence, John Wiley and Sons,##Ltd., 2005.##[10] A. Esmin, A. Aoki, and R. G. LambertTorres,Particle swarm optimization for fuzzy mem##bership functions optimization, IEEE International Conference on System Man Cybernatics,##3 (2002), 6{9.##[11] H. M. Feng, Particle swarm optimization learning fuzzy systems design, In Proceedings of##the ICITA 3rd International Conference on Information Technology and Applications, 1(July##4(7)) (2005), 363{366.##[12] M. Gen, K. Ida, Y. Li and E. Kubota, Solving bicriteria solid transportation problem with##fuzzy numbers by a genetic algorithm, Computer and Industrial Engineering, 29 (1995), 537{##[13] P. K. Giri, M. K. Maiti and M. Maiti, A solid transportation problem with fuzzy random costs##and constraints, International Journal of Mathematics in Operation Research, 4(6) (2012),##[14] A. Golnarkar, A. A. Alesheikh and M. R. Malek, Solving best path problem on multimodal##transportation networks with fuzzy costs, Iranian Journal of Fuzzy Systems, 7(3) (2010),##[15] J. Gottlieb and L. Paulmann, Genetic algorithms for the xed charge transportation problems##in: Proceedings of the IEEE Conference on Evolutionary Computation, ICEC, (1998), 330{##[16] K. B. Haley, The solid transportation problem, Operation Research, 11 (1962), 446{448.##[17] W. M. Hirsch and G. B. Dantzig, The xed charge transportation problem, Naval Research,##Logistics Quarterly, 15 (1968), 413{424.##[18] F. L. Hitchcock, The distribution of the product from several sources to numerous localities,##Journal of Mathematical Physics, 20 (1941), 224{230.##[19] H. J. Holland, Adaptation in natural and articial systems, University of Michigan press,##[20] F. Jimnez and J. L. Verdegay, Uncertain solid transportation problems, Fuzzy Sets and Systems,##100 Issues 13, 16 November (1998), 45{57.##[21] F. Jimnez and J. L. Verdegay, Solving fuzzy solid transportation problems by an evolutionary##algorithm based parametric approach, European Journal of Operational Research, 117 Issue##3, 16 September (1999), 485{510.##[22] J. Kennedy and R. C., Eberhart, Particle swarm optimisation, In Proceedings of the IEEE##International Joint Conference on Neural Network, IEEE Press, 4 (1995), 1942{1948.##[23] J. L. Kennington and V. E. Unger, A new branch and bound algorithm for the xed charge##transportation problem, Management Sciences, 22 (1976), 1116{1126.##[24] P. Kundu, S. Kar and M. Maiti, Multiobjective multiitem solid transportation problem in##fuzzy environment Appl. Math. Model., 37 (2012), 2028{2038.##[25] P. Kundu, S. Kar and M. Maiti,Fixed charge transportation problem with type2 fuzzy vari##ables, Information Sciences, 255 (2014), 170{186. ##[26] M. Last and S. Eyal,A fuzzybased lifetime extension of genetic algorithms, Fuzzy Sets and##Systems, 149 (2005), 1311{1147.##[27] J. J. Liang, A. K. Qin, P. N. Suganthan and S. Baskar, Comprehensive learning particle swarm##optimizer for global optimization of multimodal functions, IEEE Transactions on Evolutionary##Computation, 10 (June (3)) (2006), 281{295.##[28] B. Liu and Y. K. Liu, Expected value of the fuzzy variable and fuzzy expected value models,##IEEE Transactions on Fuzzy Systems, 10 (2002), 445{450.##[29] B. Liu, Theory and practice of uncertain programming, PhysicaVerlag, Heidelberg, 2002.##[30] S. Liu, Fuzzy total transportation cost measures for fuzzy solid transportation problem, Applied##Mathematics and Computation, 174 (2006), 927{941.##[31] B. Liu and K. Iwamura, A note on chance constrained programming with fuzzy coecients,##Fuzzy Sets and Systems, 100 (1998), 229{233.##[32] Z. Michalewicz,Genetic Algorithms + data structures= evolution programs, SpringerVerlag,##AI Series, New York, 1992.##[33] S. MollaAlizadehZavardehi, S. Sadi Nezhadb, R. TavakkoliMoghaddamc and M. Yazdani,##Solving a fuzzy xed charge solid transportation problem by metaheuristics, Mathematics and##Computer Modelling, 57 (2013), 1543{1558.##[34] H. NezmabadiPour, S. Yazdani, M. M. Farsangi and M. Neyestani, A solution to an economic##dispatch problem by fuzzy adaptive genetic algorithm, Iranian Journal of Fuzzy Systems, 8(3)##(2011), 1{21.##[35] A. Ojha, B. Das, S. Mondal and M. Maiti, An entropy based solid transportation problem for##general fuzzy costs and time with fuzzy equality, Mathematics and Computer Modelling, 50##(2009), 166{178.##[36] A. Ojha, B. Das, S. Mondal and M. Maiti, A Solid Transportation Problem for an item##with xed charge vehicle cost and price discounted varying charge using Genetic Algorithm,##Applied Soft Computing, 10 (2010), 100{110.##[37] A. Ojha, B. Das, S. Mondal and M. Maiti Transportation policies for single and multiobjective##transportation problem using fuzzy logic, Mathematics Computer Modelling, 53##(2011), 16371646.##[38] I. M. Oliver, D. J. Smith and J. R. C. Holland, A study of permutation crossover operators##on the travelling salesman problem, In: Proceedings of the Second International Conference##on Genetic Algorithms (ICGA'87), Massachusetts Institute of Technology, Cambridge, MA,##(1987), 224{230.##[39] E. D. Schell, Distribution of a product by several properties, In: Proceedings of 2nd Symposium##in Linear Programming, DCS/comptroller, HQ US Air Force, Washington DC, (1955),##[40] A. Sengupta and T. K. Pal, Fuzzy preference ordering of interval numbers in decision problems,##Berlin: Springer, 2009.##[41] M. Sun, J. E. Aronson, P. G. Mckeown and D. Dennis,A tabu search heuristic procedure for##xed charge transportation problem, European Journal of Operation Research, 106 (1998),##[42] K. P. Wang, L. Huang, C. G. Zhou and W. Pang, Particle swarm optimization for travelling##salesman problem, In Proc. International Conference on Machine Learning and Cybernetics,##November (2003), 15831585.##[43] X. Yan, C. Zhang, W. Luo, W. Li, W. Chen and H. Liu, Solve travelling salesman prob##lem using particle swarm optimization algorithm, International Journal of Computer Science##Issues, 9(6(2)) (2012), 264{271.##[44] L. Yang and L. Liu, Fuzzy xed charge solid transportation problem and algorithm, Applid##Soft Computing, 7 (2007), 879{889.##[45] L. A. Zadeh, Fuzzy Set as a basis for a theory of possibility, Fuzzy Sets and Systems, 1##(1978), 3{28. ##]
Derived fuzzy importance of attributes based on the weakest triangular normbased fuzzy arithmetic and applications to the hotel services
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The correlation between the performance of attributes and the overallsatisfaction such as they are perceived by the customers is often used tocalculate the importance of attributes in the crisp case. Recently, the methodwas extended, based on the standard Zadeh extension principle, to the fuzzycase, taking into account the specificity of the human thinking. Thedifficulties of calculation are important and only approximations of theanalytic results can be obtained. In the present paper we give a simplifiedand exact method to compute the derived importance of the attributes in thecase of input data given by triangular fuzzy numbers. The effectivecalculation is based on the $T_{W}$extension principle and it uses reasonablecomputer resources even if a large number of attributes and customers isconsidered. The proposed derived method is later on compared with othermethods of calculation of the fuzzy importance of attributes. The results ofa survey with respect to the quality of hotel services in Oradea (Romania)are subject to the application of the proposed method.
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85


Adrian I.
Ban
Department of Mathematics and Informatics, University of Oradea,
Universitatii 1, Oradea , Romania
Department of Mathematics and Informatics,
Romania


Olimpia I.
Ban
Department of Economics, University of Oradea, Universitatii 1,
Oradea , Romania
Department of Economics, University of Oradea,
Romania


Delia A.
Tuse
Department of Mathematics and Informatics, University of Oradea,
Universitatii 1, Oradea , Romania
Department of Mathematics and Informatics,
Romania
Triangular fuzzy number
Correlation coefficient
Importance of attributes
Performance of attributes
Hotel services
[[1] J. Abalo, J. Varela and V. Manzano, Importance values for importanceperformance analysis:##A formula for spreading out values derived from preference rankings, Journal of Bussiness##Research, 60 (2007), 115{121.##[2] D. R. Bacon, A comparison of approaches to ImportancePerformance Analysis, International##Journal of Marketing Research, 45 (2003), 55{71.##[3] A. I. Ban and L. Coroianu, Simplifying the search for eective ranking of fuzzy numbers,##IEEE Transactions on Fuzzy Systems, 23 (2014), 327{339.##[4] A. I. Ban, O. I. Ban and D. A. Tuse, Calculation of the fuzzy importance of attributes based##on the correlation coecient, applied to the quality of hotel services, Journal of Intelligent##and Fuzzy Systems, 30 (2015), 583{596.##[5] A. Ban and O. Ban, Optimization and extensions of a fuzzy multicriteria decision making##method and applications to selection of touristic destinations, Expert Systems with Applica##tions, 39 (2012), 7216{7225.##[6] A. I. Ban and L. Coroianu, Characterization of the ranking indices of triangular fuzzy num##bers. In: A. Laurent, O. Strauss, B. BouchonMeunier and R.R. Yager (Eds.), Communi##cations in Computer and Information Science, vol. 443, SpringerVerlag, Berlin, Heidelberg,##2014, pp. 254{263.##[7] O. Ban, Fuzzy multicriteria decision making method applied to selection of the best touristic##destinations, International Journal of Mathematical Models and Methods in Applied Science,##5 (2011), 264{271.##[8] O. I. Ban and I. T. Mester, Using Kano two dimensional service quality classication and##characteristic analysis from the perspective of hotels' clients of Oradea, Journal of Tourism##Studies and Research in Tourism, 18 (2014), 30{36.##[9] S. Chanas, On the interval approximation of a fuzzy number, Fuzzy Sets and Systems, 122##(2001), 353{356.##[10] P. T. Chang, P. F. Pai, K. P. Lin and M. S. 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Hong, Fuzzy measures for a correlation coecient of fuzzy numbers under TW (the##weakest tnorm)based fuzzy arithmetic operations, Information Sciences, 176 (2006), 150##[23] D. H. Hong, Shape preserving multiplications of fuzzy intervals, Fuzzy Sets and Systems, 123##(2001), 8184.##[24] D. H. Hong, On shapepreserving additions of fuzzy intervals, Journal of Mathematical Anal##ysis and Applications, 267 (2002), 369{376.##[25] E. P. Klement, R. Mesiar and E. Pap, Triangular Norms, Springer, Dordrecht, 2000.##[26] A. Kolesarova, Additive preserving the linearity of fuzzy interval, Tatra Montains Mathemat##ical Publications, 6 (1994), 75{81.##[27] M. Kumar, Applying weakest tnorm based approximate intuitionistic fuzzy arithmetic oper##ations on dierent types of intuitionistic fuzzy numbers to evaluate reliability of PCBA fault,##Applied Soft Computing, 23 (2014), 387{406.##[28] Q. 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A HyersUlamRassias stability result for functional equations in Intuitionistic Fuzzy Banach spaces
2
2
HyersUlamRassias stability have been studied in the contexts of several areas of mathematics. The concept of fuzziness and its extensions have been introduced to almost all branches of mathematics in recent times.Here we define the cubic functional equation in 2variables and establish that HyersUlamRassias stability holds for such equations in intuitionistic fuzzy Banach spaces.
1

87
96


Nabin
Chandra Kayal
Department of Mathematics, Indian Institute Of Engineering
Science and Technology, Shibpur, Howrah  711103, West Bengal, India
Department of Mathematics, Indian Institute
India


Tapas Kumar
Samanta
Department of Mathematics, Uluberia College, Uluberia,
Howrah  711315, West Bengal, India
Department of Mathematics, Uluberia College,
India


Parbati
Saha
Department of Mathematics, Indian Institute Of Engineering Science
and Technology, Shibpur, Howrah  711103, West Bengal, India
Department of Mathematics, Indian Institute
India


Binayak S.
Choudhury
Department of Mathematics, Indian Institute Of Engineering
Science and Technology, Shibpur, Howrah  711103, West Bengal, India
Department of Mathematics, Indian Institute
India
binayak12@yahoo.co.in
Cubic functional equations
tnorm
tconorm
Intuitionistic fuzzy Banach space
HyersUlamRassias stability
[[1] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87{96.##[2] J. H. Bae and W. G. Park, A xedpoint approach to the stability of a functional equation##on quadratic forms, Journal of Inequalities and Applications, 82 (2011), 1{7.##[3] J. H. Bae, W. G. Park,On the Ulam stability of the CauchyJensen equation and the additive##quadratic equation, J. Nonlinear Sci. Appl. 8(5) (2015), 710{718.##[4] G. Deschrijver, C. Cornelis and E. E. Kerre, On the representation of intuitionistic fuzzy##tnorms and tconorms, IEEE Transaction on Fuzzy Systems, 12 (2004), 45{61.##[5] G. Deschrijver and E. E. Kerre, On the relationship between some extensions of fuzzy set##theory, Fuzzy Sets and Systems, 23 (2003), 227{235.##[6] Y. Dong, On approximate isometries and application to stability of a function, J. Math.##Anal. Appl., 426(2) (2015), 125{137.##[7] A. Grabiec, The generalized HyersUlam stability of a class of functional equations, Publ.##Math. Debrecen, 48 (1996), 217{235.##[8] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A.,##27 (1941), 222{224.##[9] K. W. Jun and H. M. Kim The generalized HyersUlamRassias stability of a cubic functional##equation, J. Math. Anal. Appl., 274 (2002), 867{878. ##[10] S. M. Jung, HyersUlam stability of linear dierential equations of rst order, II, App. Math.##Lett., 19 (2006), 854{858.##[11] N. C. Kayal, P. Mondal and T. K. Samanta, The generalized Hyers  Ulam  Rassias stability##of a quadratic functional equation in fuzzy banach spaces, Journal of New Results in Science,##1(5) (2014), 83{95.##[12] N. C. Kayal, P. Mondal and T. K. Samanta, The fuzzy stability of a pexiderized functional##equation, Mathematica Moravica, 18(2) (2014), 1{14.##[13] N. C. Kayal, P. Mondal and T. K. Samanta, Intuitionistic fuzzy stability of a quadratic##functional equation, Tbilisi Mathematical Journal, 8(2) (2015), 139{147.##[14] S. O. Kim, A. Bodaghi and C. Park, Stability of functional inequalities associated with the##CauchyJensen additive functional equalities in nonArchimedean Banach spaces, J. Nonlin##ear Sci. Appl., 8(5) (2015), 776{786.##[15] Y. Lan and Y. Shen, The general solution of a quadratic functional equation and Ulam##stability, J. Nonlinear Sci. Appl. 8(5) (2015), 640{649.##[16] A. K. Mirmostafaee and M. S. Moslehian, Fuzzy versions of HyersUlamRassias theorem,##Fuzzy Sets and Systems, 159 (2008), 720{729.##[17] P. Mondal, N. C. Kayal and T. K. Samanta, The stability of pexider type functional equation##in intuitionistic fuzzy Banach spaces via xed point technique, Journal of Hyperstructures,##4(1) (2015), 37{49.##[18] A. Najati, The generalized HyersUlamRassias stability of a cubic functional equation, Turk##J Math, 31 (2007), 395{408.##[19] C. Park, Fuzzy stability of a functional equation associated with inner product space, Fuzzy##Sets and Systems, 160 (2009), 1632{1642.##[20] J. H. Park, Intuitionistic fuzzy metric spaces, Chaos, Solitons and Fractals, 22 (2004), 1039{##[21] Th. M. Rassias, On the stability of the linear mapping in Banach space, Proc. Amer. Math##ematical Society, 72(2) (1978), 297{300.##[22] R. Saadati and J. H. Park, On the intuitionistic fuzzy topological spaces, Chaos, Solitons and##Fractals, 27 (2006), 331{344.##[23] T. K. Samanta and Iqbal H. Jebril, Finite dimentional intuitionistic fuzzy normed linear##space, Int. J. Open Problems Compt. Math., 2(4) (2009), 574{591.##[24] T. K. Samanta, N. C.Kayal and P. Mondal, The Stability of a General Quadratic Functional##Equation in Fuzzy Banach Space, Journal of Hyperstructures, 1(2) (2012), 71{87.##[25] T. K. Samanta, P. Mondal and N. C. Kayal, The generalized HyersUlamRassias stability of##a quadratic functional equation in fuzzy Banach spaces, Annals of Fuzzy Mathematics and##Informatics, 6(2) (2013), 285{294.##[26] S. Shakeri, Intutionistic fuzzy stability of Jenson type mapping, J. Non linear Sc. Appl., 2(2)##(2009), 105{112.##[27] S. M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, New York,##1964(Chapter VI, Some Questions in Analysis: x1, Stability).##[28] L. A. Zadeh, Fuzzy sets, Information and control, 8 (1965), 338{353.##]
MULTIATTRIBUTE DECISION MAKING METHOD BASED ON BONFERRONI MEAN OPERATOR and possibility degree OF INTERVAL TYPE2 TRAPEZOIDAL FUZZY SETS
2
2
This paper proposes a new approach based on Bonferroni mean operator and possibility degree to solve fuzzy multiattribute decision making (FMADM) problems in which the attribute value takes the form of interval type2 fuzzy numbers. We introduce the concepts of interval possibility mean value and present a new method for calculating the possibility degree of two interval trapezoidal type2 fuzzy sets (IT2 TrFSs). Then, we develop two aggregation techniques, which are called the interval type2 trapezoidal fuzzy Bonferroni mean (IT2TFBM) operator and the interval type2 trapezoidal fuzzy weighted Bonferroni mean (IT2TFWBM) operator. We study their properties and discuss their special cases. Based on the IT2TFWBM operator and the possibility degree, a new method of multiattribute decision making with interval type2 trapezoidal fuzzy information is proposed. Finally, an illustrative example is given to verify the developed approaches and to demonstrate their practicality and effectiveness.
1

97
115


Yanbing
Gong
Department of Information Management, Hohai University,Changzhou,
Jiangsu Province, China
Department of Information Management, Hohai
China


Liangliang
Dai
Department of Information Management, Hohai University,Changzhou,
Jiangsu Province, China
Department of Information Management, Hohai
China


Na
Hu
Department of Information Management, Hohai University,Changzhou, Jiangsu
Province, China
Department of Information Management, Hohai
China
Multiattributes group decision making
Interval type2 fuzzy sets
Bonferroni mean operator
IT2TFWBM operator
[[1] C. Bonferroni, Sulle medie multiple di potenze, Bollettino dell'Unione Matematica Italiana,##5(3) (1950), 267270.##[2] G. Bortolan and R. Degani,A review of some methods for ranking fuzzy subsets, Fuzzy sets##and Systems,15(1) (1985), 119.##[3] Q. W. Cao and J. Wu, The extended COWG operators and their application to multiple##attributive group decision making problems with interval numbers, Applied Mathematical##Modelling, 35(5) (2011), 20752086.##[4] C. T. Chen, Extensions of the TOPSIS for group decisionmaking under fuzzy environment,##Fuzzy Sets and Systems, 114(1) (2000), 19.##[5] S. M. Chen, Fuzzy group decision making for evaluating the rate of aggregative risk in software##development, Fuzzy Sets and Systems, 118(1) (2001), 7588.##[6] S. M. Chen and L. W. Lee,Fuzzy multiple attributes group decisionmaking based on the##ranking values and the arithmetic operations of interval type2 fuzzy sets, Expert Systems##with Applications, 37(4) (2010), 824833. ##[7] S. M. Chen and L. W. Lee,Fuzzy multiple attributes group decisionmaking based on the##interval type2 TOPSIS method, Expert Systems with Applications, 37(4) (2010), 27902798.##[8] S. M. Chen and M. W. Yang,Fuzzy multiple attributes group decisionmaking based on ranking##interval type2 fuzzy sets, Expert Systems with Applications, 39(5) (2012), 52955308.##[9] T. Y. Chen and C. H. Chang, The extended QUALIFLEX method for multiple criteria##decision analysis based on interval type2 fuzzy sets and applications to medical decision##making, European Journal of Operational Research, 226(3) (2013), 615625.##[10] S. M. Chen and L. W. Lee,Fuzzy multiple criteria hierarchical group decision making based##on interval type2 fuzzy sets, IEEE Transactions on Systems, Man and Cybernetics, Part A:##Systems and Humans, 40(5) (2010) , 11201128.##[11] C. Carlsson and R. Fuller,On possibilistic mean value and variance of fuzzy numbers, Fuzzy##Sets and Systems, 122122(2) (2001), 315326.##[12] F.J. Cabrerizo, R. Heradio, I. J. Prez and E. HerreraViedma,A selection process based on##additive consistency to deal with incomplete fuzzy linguistic information, Journal of Universal##Computer Science, 16(1) (2010), 6281.##[13] Y. C. Dong, G. Q. Zhang, W. C. Hong and Y. F. Xu, Consensus models for AHP group##decision making under row geometric mean prioritization method, Decision Support Systems,##49(3) (2010), 281{289.##[14] Z. P. Fan and Y. Liu, A method for group decisionmaking based on multigranularity uncer##tain linguistic information, Expert Systems with Applications,37(5) (2010), 40004008.##[15] Y. B. Gong, N. Hu, J. G. Zhang, G. F. Liu and J. G. Deng, Multiattribute group decision##making method based on geometric Bonferroni mean operator of trapezoidal interval type2##fuzzy numbers, Computers and Industrial Engineering, 81(3) (2015), 167176.##[16] J. H. Hu, Y. Zhang, X. H. Chen and Y. M. Liu, Multicriteria decision making method##based on possibility degree of interval type2 fuzzy number, KnowledgeBased Systems, 43(5)##(2013), 2129.##[17] P. D. Liu, A weighted aggregation operators multiattribute group decisionmaking method##based on intervalvalued trapezoidal fuzzy numbers, Expert Systems with Applications, 38(1)##(2011), 10531060.##[18] D. F. Li, A fuzzy closeness approach to fuzzy multiattribute decision making, Fuzzy Opti##mization and Decision Making,6(3) (2007), 237254.##[19] C. J. Lin and W. Wu, A causal analytical method for group decisionmaking under fuzzy##environment, Expert Systems with Applications,34(1) (2008), 205213.##[20] D. F. Li, A new methodology for fuzzy multiattribute group decision making with multi##granularity and nonhomogeneous information, Fuzzy Optimization and Decision Making,##9(1) (2010), 83103.##[21] P. D. Liu and F. Jin,The trapezoid fuzzy linguistic Bonferroni mean operators and their##application to multiple attribute decision making, Scientia Iranica, 19(6) (2012),19471959.##[22] D. Q. Li, W. Y. Zeng and J. H. Li, Note on uncertain linguistic Bonferroni mean operators##and their application to multiple attribute decision making, Applied Mathematical Modelling,##392) (2015), 894900.##[23] J. M. Mendel, R. I. John and F. Liu, Interval type2 fuzzy logic systems made simple, IEEE##Transactions on Fuzzy Systems, 14(6) (2006), 808821.##[24] J. M. Mendel, Uncertain rulebased fuzzy logic systems: introduction and new directions,##Upper Saddle River, PrenticeHall, NJ, 2001.##[25] J. M. Mendel and H. W. Wu, Type2 fuzzistics for symmetric interval type2 fuzzy sets: part##1, forward problems, IEEE Transactions on Fuzzy Systems,14(6) (2006), 781792.##[26] J. M. Mendel and H. W. Wu, Type2 fuzzistics for symmetric interval type2 fuzzy sets: part##2,inverse problems, IEEE Transactions on Fuzzy Systems,15(2) (2007), 301308.##[27] M. J. Tsai and C. S. Wang, The extended COWG operators and their application to multi##ple attributive group decision making problems with interval numbers, Expert Systems with##Applications, 34(4) (2008), 29212936.##[28] G. W. Wei and X. F. Zhao, Some dependent aggregation operators with 2tuple linguistic in##formation and their application to multiple attribute group decision making , Expert Systems##with Applications, 39(5) (2012),58815886.##[29] J. Wang and Y. I. Lin, Some dependent aggregation operators with 2tuple linguistic infor##mation and their application to multiple attribute group decision making , Fuzzy Sets and##Systems, 134(3) (2003),343363.##[30] Z. Wu and Y. Chen, The maximizing deviation method for group multiple attribute decision##making under linguistic environment, Educational Philosophy and Theory, 158(14) (2007),##16081617.##[31] D. R. Wu and J. M. Mendel, Aggregation using the linguistic weighted average and interval##type2 fuzzy sets, IEEE Transactions on Fuzzy Systems, 15(3) (2007), 11451161.##[32] D. R. Wu and J. M. Mendel, A vector similarity measure for linguistic approximation: in##terval type2 and type1 fuzzy sets, Information Sciences, 178(2) (2008), 381402.##[33] D. R. Wu and J. M. Mendel, A comparative study of ranking methods, similarity measures##and uncertainty measures for interval type2 fuzzy sets, Information Sciences, 179(8) (2009),##11691192.##[34] D. R. Wu and J. M. Mendel,Corrections to aggregation using the linguistic weighted average##and interval type2 fuzzy sets, IEEE Transactions on Fuzzy Systems, 15(6) (2008), 11451161.##[35] W. Z. Wang and X. W. Liu,Multiattribute group decision making models under interval##type2 fuzzy environment, KnowledgeBased Systems, 30(6) (2012), 121128.##[36] G. W. Wei, X. F. Zhao, R. Lin and H. J. Wang, Uncertain linguistic Bonferroni mean##operators and their application to multiple attribute decision making, Applied Mathematical##Modelling, 37(7) (2013), 52775285.##[37] Z. S. Xu, An interactive procedure for linguistic multiple attribute decision making with##incomplete weight information, Fuzzy Optimization and Decision Making,6(1) (2007), 17##[38] Z. S. Xu and R. R. Yager,Intuitionistic fuzzy Bonferroni means, IEEE Transactions on Sys##tems, Man, and Cybernetics  Part B, 41(2) (2011), 568578.##[39] M. M. Xia, Z.S. Xu and B. Zhu, Geometric Bonferroni means with their application in##multicriteria decision making, Technical Report, 2011.##[40] M. M. Xia, Z. S. Xu and B. Zhu,Geometric Bonferroni means with their application in##multicriteria decision making, KnowledgeBased Systems, 40(1) (2013), 88100.##[41] R. R. Yager, On generalized Bonferroni mean operators for multicriteria aggregation, Inter##national Journal of Approximate Reasoning, 50(8) (2009), 12791286.##[42] L. Zadeh, The concept of a linguistic variable and its application to approximate reasoning,##Part 1, Information Sciences, 8(3) (1975), 199249.##[43] Z. M. Zhang and S. H. Zhang,A novel approach to multi attribute group decision making##based on trapezoidal interval type2 fuzzy soft sets, Applied Mathematical Modeling, 37(7)##(2013), 49484971.##[44] B. Zhu, Z. S. Xu and M. M. Xia,Hesitant fuzzy geometric Bonferroni means, Information##Sciences, 205 205(1) (2012), 7285.##]
Triangular Intuitionistic Fuzzy Triple Bonferroni Harmonic Mean Operators and Application to Multiattribute Group Decision Making
2
2
As an special intuitionistic fuzzy set defined on the real number set, triangular intuitionistic fuzzy number (TIFN) is a fundamental tool for quantifying an illknown quantity. In order to model the decision maker's overall preference with mandatory requirements, it is necessary to develop some Bonferroni harmonic mean operators for TIFNs which can be used to effectively intergrate the information of attribute values for multiattribute group decision making (MAGDM) with TIFNs. The purpose of this paper is to develop some Bonferroni harmonic operators of TIFNs and apply to the MAGDM problems with TIFNs. The weighted possibility means of TIFN are firstly defined. Hereby, a new lexicographic approach is presented to rank TIFNs sufficiently considering the risk preference of decision maker. The sensitivity analysis on the risk preference parameter is made. Then, three kinds of triangular intuitionistic fuzzy Bonferroni harmonic aggregation operators are defined, including a triangular intuitionistic fuzzy triple weighted Bonferroni harmonic mean operator (TIFTWBHM) operator, a triangular intuitionistic fuzzy triple ordered weighted Bonferroni harmonic mean (TIFTOWBHM) operator and a triangular intuitionistic fuzzy triple hybrid Bonferroni harmonic mean (TIFTHBHM) operator. Some desirable properties for these operators are discussed in detail. By using the TIFTWBHM operator, we can obtain the individual overall attribute values of alternatives, which are further integrated into the collective ones by the TIFTHBHM operator. The ranking order of alternatives is generated according to the collective overall attribute values of alternatives. A real investment selection case study verifies the validity and applicability of the proposed method.
1

117
145


ShuPing
Wan
College of Information Technology, Jiangxi University of Finance
and Economics, Nanchang 330013, China
College of Information Technology, Jiangxi
China


YongJun
Zhu
College of Information Technology, Jiangxi University of Finance
and Economics, Nanchang 330013, China
College of Information Technology, Jiangxi
China
Multiattribute group decision making
Triangular intuitionistic fuzzy number
Possibility mean
Bonferroni mean
Harmonic mean
[[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1) (1986), 87{96.##[2] G. Beliakov, S. James, J. Mordelova, T. Ruckschlossova and R. Yager, Generalized Bon##ferroni mean operators in multicriteria aggregation, Fuzzy Sets and Systems, 161 (2010),##2227{2242.##[3] C. Bonferroni, Sulle medie multiple di potenze, Bolletino Matematica Italiana, 5 (1950),##[4] C. Carlsson and R. Fuller, On possibilistic mean value and deviation of fuzzy numbers,##Fuzzy Sets and Systems, 122 (2001), 315326.##[5] H. Y. Chen, C. L. Liu and Z. H. Sheng, Induced ordered weighted harmonic averaging##(IOWHA) operator and its application to combination forecasting method, Chinese Journal##of Management Science, 12(5) (2004), 3540.##[6] J. Y. Dong and S. P. Wan, A new method for multiattribute group decision making with##triangular intuitionistic fuzzy numbers, Kybernetes, 45(1) (2016), 158180.##[7] J. Y. Dong and S. P.Wan, A new method for prioritized multicriteria group decision mak##ing with triangular intuitionistic fuzzy numbers, Journal of intelligent and Fuzzy systems,##30 (2016), 17191733.##[8] B. Dutta and D. Guha, Trapezoidal intuitionistic fuzzy Bonferroni means and its applia##tion in multiattribute decision making, Fuzzy Systems (FUZZ), (2013) IEEE International##Conference on. IEEE, (2013), 18.##[9] D. P. Filev and R. R. Yager, On the issue of obtaining OWA operator weights, Fuzzy Sets##and Systems, 94 (2) (1998), 157169.##[10] R. Fuller and P. Majlender, On weighted possibilistic mean and variance of fuzzy numbers,##Fuzzy Sets and Systems, 136 (2003), 363374.##[11] D. F. Li, A note on "using intuitionistic fuzzy sets for faulttree analysis on printed circuit##board assembly", Microelectronics Reliability, 48(10) (2008), 1741. ##[12] D. F. Li, A ratio ranking method of triangular intuitionistic fuzzy numbers and its appli##cation to MADM problems, Computers and Mathematics with Applications, 60 (2010),##15571570.##[13] D. F. Li, J. X. Nan and M. J. Zhang, A ranking method of triangular intuitionistic fuzzy##numbers and application to decision making, International Journal of Computational Intelligence##Systems, 3(5) (2010), 522530.##[14] J. X. Nan, D. F. Li and M. J. Zhang, A lexicographic method for matrix games with##payos of triangular intuitionistic fuzzy numbers, International Journal of Computational##Intelligence Systems, 3(3) (2010), 280289.##[15] J. H. Park and E. J. Park, Generalized fuzzy Bonferroni harmonic mean operators##and their applications in group decision making, Journal of Applied Mathematics(2013),##http://dx.doi.org/10.1155/2013/604029.##[16] M. H. Shu, C. H. Cheng and J. R. Chang, Using intuitionistic fuzzy sets for fault tree##analysis on printed circuit board assembly, Microelectronics Reliability, 46(12) (2006),##21392148.##[17] H. Sun and M. Sun, Generalized Bonferroni harmonic mean operators and their applica##tion to multiple attribute decision making, Journal of Computational Information Systems,##8 (2012), 57175724.##[18] S. P. Wan, G. L. Xu, F. Wang and J. Y. Dong, A new method for Atanassov's interval##valued intuitionistic fuzzy MAGDM with incomplete attribute weight information, Information##Sciences, 316 (2015), 329347.##[19] S. P. Wan and D. F. Li, Fuzzy mathematical programming approach to heterogeneous##multiattribute decisionmaking with intervalvalued intuitionistic fuzzy truth degrees, Information##Sciences, 325 (2015), 484503.##[20] S. P. Wan and J. Y. Dong, Intervalvalued intuitionistic fuzzy mathematical programming##method for hybrid multicriteria group decision making with intervalvalued intuitionistic##fuzzy truth degrees, Information Fusion, 26 (2015), 4965.##[21] S. P. Wan and J. Y. Dong, Power geometric operators of trapezoidal intuitionistic fuzzy##numbers and application to multiattribute group decision making, Applied Soft Computing,##29 (2015), 153168.##[22] S. P. Wan, F. Wang and J. Y. Dong,A novel group decision making method with intuition##istic fuzzy preference relations for RFID technology selection, Applied Soft Computing,##38 (2016), 405422.##[23] S. P. Wan, F. Wang, L. L. Lin and J. Y. Dong,An intuitionistic fuzzy linear programming##method for logistics outsourcing provider selection, KnowledgeBased Systems, 82 (2015),##[24] S. P. Wan, F. Wang and J. Y. Dong,A novel risk attitudinal ranking method for intu##itionistic fuzzy values and application to MADM, Applied Soft Computing, 40 (2016),##[25] S. P. 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On the multivariate process capability vector in fuzzy environment
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The production of a process is expected to meet customer demands, specifications or engineering tolerances. The ability of a process to meet these expectations is expresed as a single number using a process capability index. When the quality of the products relates to more than one characteristic, multivariate process capability indices are applied. As it is known, in some circumstances we are faced with imprecise data. So, fuzzy logic is engaged to deal with them. In this article, the specification limits and the target value of each characteristic and also, the data gathered from the process are assumed to be imprecise and a new fuzzy multivariate capability vector is introduced. As a whole, the present article provides a research of the application of fuzzy logic in multivariate capability vector.
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Zainab
Abbasi Ganji
Department of Statistics, Faculty of Mathematical Sciences,
Ferdowsi University of Mashhad, Mashhad, Iran
Department of Statistics, Faculty of Mathematical
Iran
abbasiganji@stu.um.ac.ir


Bahram
Sadeghpour Gildeh
Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran
Department of Statistics, Faculty of Mathematical
Iran
sa deg hpour@umz.ac.ir
Multivariate normal distribution
Multivariate process capability vector
fuzzy logic
Triangular fuzzy matrix
Fuzzy linear equation system
Ranking function
[[1] M. A. Basaran, Calculating fuzzy inverse matrix using fuzzy linear equation system, Applied##Soft Computing, 12 (2012), 1810{1813.##[2] K. S. Chen and W. L. Pearn, Capability indices for processes with asymmetric tolerances,##Journal of the Chinese Institute of Engineers, 24(5) (2001), 559{568.##[3] M. Dehghan, M. Ghatee and B. Hashemi, Inverse of a fuzzy matrix of fuzzy numbers, International##Journal of Computer Mathematics, 86(8) (2009), 1433{1452.##[4] P. Fortemps and M. Roubens, Ranking and defuzzification methods based on area compensation,##Fuzzy Sets and Systems, 82 (1996) 319{330.##[5] J. E. Jackson, Quality control methods for two related variables, Industrial Quality Control,##(1956), 4{8. ##[6] I. Kaya and C. Kahraman, Fuzzy process capability analyses: An application to teaching##processes, Journal of Intelligent and Fuzzy Systems, 19(45) (2008), 259{272.##[7] I. Kaya and C. Kahraman, Fuzzy robust process capability indices for risk assessment of air##pollution, Stochastic Environmental Research and Risk Assessment, 23(4) (2009), 529{541.##[8] I. Kaya and C. Kahraman, Development of fuzzy process accuracy index for decision making##problems, Information Sciences, 180(6) (2010), 861{872.##[9] A. Parchami, M. Mashinchi and H. R. Maleki, Fuzzy confidence interval for fuzzy process##capability index, Journal of Intelligent and Fuzzy Systems, 17 (2006), 287{295.##[10] A. Parchami and M. Mashinchi, Fuzzy estimation for process capability indices, Information##Sciences, 177 (2007), 1452{1462.##[11] A. Parchami, B. Sadeghpour Gildeh, M. Nourbakhsh and M. Mashinchi, A new generation##of process capability indices based on fuzzy measurements, Journal of Applied Statistics, 41##(2014), 1122{1136.##[12] W. L. Pearn, S. Kotz and N. L. Johnson, Distributional and inferential properties of process##capability indices, Journal of Quality Technology, 24 (1992), 216{233.##[13] B. Sadeghpour Gildeh, Comparison of and process capability indices in the case of measurement##error occurrence, IFSA World Congress, Istanbul, Turkey, (2003), 563{567.##[14] B. Sadeghpour Gildeh, Measurement error effects on the performance of the process capability##index based on fuzzy tolerance interval, Annals of Fuzzy Mathematica and Informatics, 2##(2011), 17{32.##[15] B. Sadeghpour Gildeh and V. Moradi, Fuzzy tolerance region and process capability analysis,##Advances in Intelligent and Soft Computing, 157 (2012), 183{193.##[16] H. Shahriari and M. Abdollahzadeh, A new multivariate process capability vector, Quality##Engineering, 21(3) (2009), 290{299.##[17] J. J. H. Shiau, C. L. Yen, W. L. Pearn and W. T. Lee, YieldRelated process capability##indices for processes of multiple quality characteristics, Quality and Reliability Engineering##International, 29 (2013), 487{507.##[18] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338{353.##[19] M. Zhang, G. A. Wang, H. E. Shuguang and H. E. Zhen, Modified multivariate process capability##index using principal component analysis, Chinese Journal of Mechanical Engineering,##27(2) (2014), 249{259.##]
Cartesianclosedness of the category of $L$fuzzy Qconvergence spaces
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The definition of $L$fuzzy Qconvergence spaces is presented by Pang and Fang in 2011. However, Cartesianclosedness of the category of $L$fuzzy Qconvergence spaces is not investigated. This paper focuses on Cartesianclosedness of the category of $L$fuzzy Qconvergence spaces, and it is shown that the category $L$$mathbf{QFCS}$ of $L$fuzzy Qconvergence spaces is Cartesianclosed.
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168


Juan
Li
School of Mathematics, Beijing Institute of Technology, Beijing 100081,
PR China
School of Mathematics, Beijing Institute
China
lijuan201209@sohu.com
$L$fuzzy filter
$L$fuzzy Qconvergence space
$L$fuzzy topology
Cartesianclosedness
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Persiantranslation vol. 13, no. 5, October 2016
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