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SOME RESULTS OF MOMENTS OF UNCERTAIN RANDOM VARIABLES
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Chance theory is a mathematical methodology for dealing with indeterminatephenomena including uncertainty and randomness.Consequently, uncertain random variable is developed to describe the phenomena which involveuncertainty and randomness.Thus, uncertain random variable is a fundamental concept in chance theory.This paper provides some practical quantities to describe uncertain random variable.The typical one is the expected value, which is the uncertain version of thecenter of gravity of a physical body.Mathematically, expectations are integrals with respect to chance distributionsor chance measures.In fact, expected values measure the center of gravity of a distribution; they aremeasures of location. In order to describe a distribution in brief terms thereexist additional measures, such as the variance which measures the dispersionor spread, and moments.For calculating the moments of uncertain random variable, some formulas are provided through chance distribution and inverse chance distribution. The main results are explained by using several examples.
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Hamed
Ahmadzade
Department of Statistics, University of Sistan and Baluchestan,
Zahedan, Iran
Department of Statistics, University of Sistan
Iran
ahmadzadeh.h.63@gmail.com


Yuhong
Sheng
College of Mathematical and System Sciences, Xinjiang University,
Urumqi 830046, China
College of Mathematical and System Sciences,
China
shengyuhong@sina.com


Fatemeh
Hassantabar Darzi
Department of Statistics, University of Sistan and
Baluchestan, Zahedan, Iran
Department of Statistics, University of Sistan
Iran
Chance theory
Uncertain random variable
Chance distribution
Moments
[[1] X. Chen and W. Dai, Maximum entropy principlefor uncertain variables, International Jour##nal of Fuzzy Systems, 13(3) (2011), 232{236.##[2] X. Chen, S. Kar and D. Ralescu, Crossentropy measure of uncertain variables, Information##Sciences, 201 (2012), 53{60.##[3] X. Chen and D. Ralescu, Liu process and uncertain calculus, Journal of Uncertainty Analysis##and Applications, 1(3) (2013), 1{ 12.##[4] W. Dai and X. Chen, Entropy of function of uncertain variables, Mathematics and Computer##Modelling, 55 (2012), 754{760.##[5] H. Y. Guo and X. S. Wang, Variance of uncertain random variables, Journal of Uncertainty##Analysis and Applications, 2(6) (2014), 1{7.##[6] Y. C. Hou, Subadditivity of chance measur, Journal of Uncertainty Analysis and Applications,##2(14) (2014), 1{8. ##[7] A. N. Kolmogorov, Grundbegrie der Wahrscheinlichkeitsrechnung, Julius Springer, Berlin,##[8] R. Kruse and K. Meyer, Statistics with Vague Data, Reidel Publishing Company, Dordrecht,##[9] B. Liu, Uncertainty Theory, 5th ed., http://orsc.edu.cn/liu/ut.pdf. 2014.##[10] B. Liu, Uncertainty Theory, 2nd ed., SpringerVerlag, Berlin, 2007.##[11] B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 31##(2009), 3{10.##[12] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty,##SpringerVerlag, Berlin, 2010.##[13] B. Liu, Toward uncertain nance theory, Journal of Uncertainty Analysis and Applications,##1(1) (2013), 1{15.##[14] Y. H. Liu, Uncertain random variables: A mixture of uncertainty and randomness, Soft##Computing, 17(4) (2013), 625{634.##[15] Y. H. Liu, Uncertain random programming with applications, Fuzzy Optimization and Deci##sion Making, 12(2) (2013), 153{169.##[16] Y. H. Liu and M. H. Ha, Expected value of function of uncertain variables, Journal of Un##certain Systems, 4(3) (2010), 181{186.##[17] Y. K. Liu and B. Liu, Fuzzy random variables: a scalar expected value operator, Fuzzy##Optimization and Decision Making, 2(2) (2003), 143{160.##[18] Y. K. Liu and B. Liu, Fuzzy random programming with equilibrium chance constraints, In##formtion Sciences 170 (2005), 363{395.##[19] Z. X. Peng and K. Iwamura, A sucient and necessary condition of uncertainty distribution,##Journal of Interdisciplinary Mathematics, 13(3) (2010), 277{285.##[20] M. Puri and D. Ralescu, Fuzzy random variables, Journal of Mathmatical Application, 114##(1986), 409{422.##[21] Y. H. Sheng and S. Kar, Some results of moments of uncertain variable through inverse##uncertainty distribution, Fuzzy Optimization and Decision Making, 14 (2015), 57{76.##[22] Y. H. Sheng and K. Yao, Some formulas of variance of uncertain random variable, Journal##of Uncertainty Analysis and Applications, 2(12) (2014), 1{10.##[23] J. L. Teugels and B. Sundt, Encyclopedia of actuarial science, Wiley & Sons, 1 (2004).##[24] M. Wen and R. Kang, Reliability analysis in uncertain random system, Fuzzy Optimization##and Decision Making, doi:10.1007/s107000169235y, (2016).##[25] K. Yao, A formula to calculate the variance of uncertain variable, Soft Computing, 19(10)##(2015), 2947{2953.##]
A JOINT DUTY CYCLE SCHEDULING AND ENERGY AWARE ROUTING APPROACH BASED ON EVOLUTIONARY GAME FOR WIRELESS SENSOR NETWORKS
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Network throughput and energy conservation are two conflicting important performance metrics for wireless sensor networks. Since these two objectives are in conflict with each other, it is difficult to achieve them simultaneously. In this paper, a joint duty cycle scheduling and energy aware routing approach is proposed based on evolutionary game theory which is called DREG. Making a tradeoff between energy conservation and network throughput, the proposed approach prolongs the network lifetime. The paper is divided into the following sections: Initially, the discussion is presented on how the sensor nodes can be scheduled to sleep or wake up in order to reduce energy consumption in idle listening. The sensor wakeup/sleep scheduling problem with multiple objectives is formulated as an evolutionary game theory. Then, the evolutionary game theory is applied to find an optimal wakeup/sleep scheduling policy, based on a tradeoff between network throughput and energy efficiency for each sensor. The evolutionary equilibrium is proposed as a solution for this game. In addition, a routing approach is adopted to propose an energy aware fuzzy logic in order to prolong the network lifetime. The results show that the proposed routing approach balances energy consumption among the sensor nodes in the network, avoiding rapid energy depletion of sensors that have less energy. The proposed simulation study shows the more efficient performance of the proposed system than other methods in term of network lifetime and throughput.
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M. S.
Kordafshari
Department of Computer Engineering, Science and Research
Branch, Islamic Azad University, Tehran, Iran
Department of Computer Engineering, Science
Iran


A.
Movaghar
Department of Computer Engineering, Sharif University of Technology, Tehran, Iran
Department of Computer Engineering, Sharif
Iran


M. R.
Meybodi
Computer Engineering and Information Technology Department,
Amirkabir University of Technology, Tehran, Iran
Computer Engineering and Information Technology
Iran
Wireless sensor network
Duty cycle scheduling
Energy aware routing
Evolutionary game theory
Distributed reinforcement learning
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Yao, Gureen Game, An energyecient QoS control scheme for wireless##sensor networks In Proceedings of 2011 International Green Computing Conference, Orlando,##FL, USA, (2011), 2528.##[7] A. Behzadan and A. Anpalagan, Prolonging network life time via nodal energy balancing in##heterogeneous wireless sensor networks, In: 2011 IEEE international conference on commu##nications, Kyoto, Japan (2011), 15.##[8] M. Buettner, G. V. Yee, E. Anderson and R. Han, XMAC: a short preamble MAC protocol##for dutycycled wireless sensor networks, In Proc. of the 4th International Conference on##Embedded Networked Sensor Systems , (2006), 307320.##[9] S. S. Chiang and C. H. Huang, A minimum hop routing protocol for home security systems##using wireless sensor networks, IEEE Transactions on Consumer Electronics, 53(4) (2007).##[10] A. M. Colman, Cooperation, psychological game theory, and limitations of rationality in##social interaction, Behavioral and Brain Sciences, 26 (2003), 139198. ##[11] J. C. Dagher, M. W. Marcellin and M. A. Neifeld, A theory for maximizing the lifetime of##sensor networks, IEEE Transaction on Communications, 55(2) (2007), 323332.##[12] D. Fudenberg and D. K. Levine, The theory of learning in games. cambridge, MIT Press,##Cambridge, MA, 1998.##[13] T. He, J. A. Stankovic, C. Lu and T. Abdelzaher, SPEED: A stateless protocol for real##time communication in sensor networks, Proceedings of IEEE International Conference on##Distributed Computing Systems, (2005), 4655.##[14] M. Javidi and L. Aliahmadipour, Application of game theory approaches in routing protocols##for wireless networks, In proceedings of 2011 International Conference on Numerical Analysis##and Applied Mathematics, Halkidiki, Greece, (2011), 1925.##[15] Z. Jia, M. Chundi and H. Jianbin, Game theoretic energy balance routing in wireless sensor##networks, In Chinese control conference, (2007), 420424.##[16] R. Kannan and S. S. Iyengar, Gametheoretic models for reliable pathlength and energy##constrained routing with data aggregation in wireless sensor networks, IEEE JSAC,##22(6)(2004), 11411150.##[17] K. Lin, T. Xu, M. M. Hassan and A. Alamri An energyeciency node scheduling game based##on task prediction in WSNs , Mobile NetwAppl, Springer Science and Business Media New##York, 20 (2015), 583592.##[18] G. Lu, B. Krishnamachari and C. S. Raghavendra, An adaptive energyecient and low##latency MAC for treebased data gathering in sensor networks, Wirel. Commun. Mob. Com##put, Published online in Wiley Inter Science., 7 (2007), 863875.##[19] R. Machado and S. Tekinay, A survey of game theoretic approaches in wireless senso rnet##works, ComputNetw, 52(16) (2008), 30473061.##[20] D. Niyato and E. Hossain, wireless sensor networks with energy harvesting technologies: a##gametheoretic approach to optimal energy management, IEEE Wireless Communications,##[21] D. Niyato and E. Hossain, Dynamics of network selection in heterogeneous wireless networks:##an evolutionary game approach, IEEE Transactions on vehicular technology, 58(4) (2009).##[22] N. A. Pantazis, S. A. Nikolidakis and D. D. Vergados, Energyecient routing protocols##in wireless sensor networks: a survey, IEEE Communications Surveys & Tutorials, 15(2)##[23] J. Polastre, J. Hill, and D. Culler, Versatile low power media access for wireless sensor##networks, In The Second ACM Conference on Embedded Networked Sensor Systems, (2004),##[24] O. Powell and A. Jarry, Gradient based routing in wireless sensor networks: a mixed strategy,##CoRR Distributed, Parallel and Cluster Computing, 2005.##[25] R. Rajagopalan and P. K. Varshney, Data aggregation techniques in sensor networks: A##survey, IEEE Commun. Surv. Tutor., 8 (2006).##[26] T. Rault, A. Bouabdallah and Y. Challal, Energyeciency in wireless sensor networks: a##topdown review approach, ComputNetw, 67 (2014), 104122.##[27] H. Ren and M. Meng, Gametheoretic modeling of joint topology control and power scheduling##for wireless heterogeneous sensor networks, IEEE Trans. Autom. Sci. Eng., 6 (2009), 610625.##[28] A. Schillings and K. Yang, VGTR A collaborative, energy and information aware routing##algorithm for wireless sensor networks through the use of game theory, In Proceedings of 3rd##International Geosensor Networks Conference, Oxford, UK, (2009), 1314.##[29] H. Shpungin and Z. Li Throughput and energy eciency in wireless AdHoc networks with##gaussian channels, IEEE Communications Society, (2010), 289298.##[30] J. M. Smith, Evolution and the Theory of Games: In situations characterized byconict of##interest, the best strategy to adopt depends on what others are doing, American Scientist,##[31] R. S. Sutton and A. G. Barto, Reinforcement learning: an introduction (adaptive computation##and machine learning), MIT Press, Cambridge, MA, 1998.##[32] D. Tudose, L. Gheorghe and N. T. Apus, Radio transceiver consumption modeling for multi##hop wireless sensor networks, UPB Scientic Bulletin, Series C, 75(1) (2013), 1726. ##[33] Y. Wu, Zh. Mao and S. Fahmy, Constructing maximumlifetime datagathering forests in##sensor networks, IEEE/ACM Transactions on Networking, 18(5) (2010).##[34] G. Yang and G. Zhang, A power control algorithm based on noncooperative game for wireless##sensor networks, In Proceedings of 2011 International Conference on Electronic & Mechanical##Engineering and Information Technology, Harbin, China, (2011), 1214.##[35] KLA. Yau, P. Komisarczuk and P. D. Teal, Reinforcement learning for context awareness##and intelligence in wireless networks: review, new features and open issues, J Netw Comput##Appl., 35(1) (2012), 253267.##[36] W. Ye, J. Heidemann, and D. Estrin, Medium access control with coordinated, adaptive##sleeping for wireless sensor networks, ACM Transactions on Networking, 12(3) (2004).##[37] L. Zhao, L. Guo, L. Cong and H. Zhang, An energyecient MAC protocol for WSNs: game##theoretic constraint optimization with multiple objectives, WirelSensNetw, (2009), 358364.##[38] M. Zheng, Game theory used for reliable routing modeling in wireless sensor networks, In##International Conference on Parallel and Distributed Computing, Applications and Technolo##gies, China, (2010), 280284.##]
MULTIOBJECTIVE ROUTING AND SCHEDULING IN FLEXIBLE MANUFACTURING SYSTEMS UNDER UNCERTAINTY
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The efficiency of transportation system management plays an important role in the planning and operation efficiency of flexible manufacturing systems. Automated Guided Vehicles (AGV) are part of diversified and advanced techniques in the field of material transportation which have many applications today and act as an intermediary between operating and storage equipment and are routed and controlled by an intelligent computer system. In this study, a twoobjective mathematical programming model is presented to integrate flow shop scheduling and routing AVGs in a flexible manufacturing system. In reallife problems parameters like demand, due dates and processing times are always uncertain. Therefore, in order to solve a realistic problem, foregoing parameters are considered as fuzzy in our proposed model. Subsequently, to solve fuzzy mathematical programming model, one of the most effective technique in the literature is used. To solve the problem studied, two metaheuristic algorithms of Nondominated Sorting Genetic AlgorithmII (NSGAII) and multiobjective particle swarm optimization (MOPSO) are offered that the accuracy of mathematical models and efficiency of algorithms provided are assessed through numerical examples.
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Ahmad
Mehrabian
Department of Industrial Engineering, SouthTehran Branch,
Islamic Azad University, Tehran, Iran
Department of Industrial Engineering, SouthTehran
Iran
ahmad.mehrabian@outlook.com


Reza
TavakkoliMoghaddam
Department of Industrial Engineering, SouthTehran
Branch, Islamic Azad University, Tehran, Iran
Department of Industrial Engineering, SouthTehran
Iran
tavakoli@ut.ac.ir


Kaveh
KhaliliDamaghani
Department of Industrial Engineering, SouthTehran Branch,
Islamic Azad University, Tehran, Iran
Department of Industrial Engineering, SouthTehran
Iran
Scheduling
Routing
Automated guided vehicle
Metaheuristic algorithm
Flexible manufacturing
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Inuiguchi, Dynamic optimization of simultaneous dispatching##ictfree routing for automated guided vehiclesPetri net decomposition approach,##Journal of Advanced Mechanical Design, Systems, and Manufacturing, 4(3) (2010), 701715.##[61] P. Udhayakumar and S. Kumanan, Task scheduling of AGV in FMS using nontraditional##optimization techniques, International Journal of Simulation Modelling, 9(1) (2010), 2839.##[62] G. Ulusoy, F. SivrikayaSerifo^glu and Bilge, U. A genetic algorithm approach to the simul##taneous scheduling of machines and automated guided vehicles, Computers & Operations##Research, 24(4) (1997), 335351.##[63] B. Vahdani, Vehicle positioning in cell manufacturing systems via robust optimization, Applied##Soft Computing, 24 (2014), 7885.##[64] M. C. Van der Heijden, M. Ebben, N. Gademann, and A. van Harten, Scheduling vehicles in##automated transportation systems Algorithms and case study, OR spectrum,24(1) (2002a),##[65] M. C. Van der Heijden, A. Van Harten, M. J. R. Ebben, Y. A. Saanen, E. C. Valentin, and##A. Verbraeck, Using simulation to design an automated underground system for transporting##freight around Schiphol Airport, Interfaces, 32(4) (2002b), 119.##[66] B. Veeravalli, G. Rajesh and N. Viswanadham, Design and analysis of optimal material##distribution policies in ##exible manufacturing systems using a single AGV, International##journal of production research, 40(12) (2002), 29372954.##[67] I. F. Vis and I. Harika, Comparison of vehicle types at an automated container terminal, OR##Spectrum, 26(1) (2004), 117143.##[68] I. F. Vis, Survey of research in the design and control of automated guided vehicle systems,##European Journal of Operational Research, 170(3) (2006), 677709.##[69] K. Vivaldini, L. F. Rocha, N. J. Martarelli, M. Becker and A. P. Moreira, Integrated tasks##assignment and routing for the estimation of the optimal number of AGVS, The International##Journal of Advanced Manufacturing Technology, 82(14) (2016), 719736.##[70] N. Wu, and M. Zhou, Modeling and deadlock control of automated guided vehicle systems,##Mechatronics, IEEE/ASME Transactions on,9(1) (2004), 5057.##[71] C. H. Yang, Y. S. Choi and T. Y. Ha, Simulationbased performance evaluation of transport##vehicles at automated container terminals, Or Spectrum,26(2) (2004), 149170.##[72] J. W. Yoo, E. S. Sim, C. Cao and J. W. Park, An algorithm for deadlock avoidance in an##AGV System, The International Journal of Advanced Manufacturing Technology, 26(56)##(2005), 659668.##[73] M. B. ZA Remba, A. Obuchowicz, Z. A. Banaszak and K. J. Jed Rzejek, A maxalgebra##approach to the robust distributed control of repetitive AGV systems, International journal##of production research, 35(10) (1997), 26672688.##]
TAUBERIAN THEOREMS FOR THE EULERNORLUND MEANCONVERGENT SEQUENCES OF FUZZY NUMBERS
2
2
Fuzzy set theory has entered into a large variety of disciplines of sciences,technology and humanities having established itself as an extremely versatileinterdisciplinary research area. Accordingly different notions of fuzzystructure have been developed such as fuzzy normed linear space, fuzzytopological vector space, fuzzy sequence space etc. While reviewing theliterature in fuzzy sequence space, we have seen that the notion of Tauberiantheorems for the EulerN"{o}rlund meanconvergent sequences of fuzzy numbershas not been developed. In the present paper, we introduce some new conceptsabout statistical convergence of sequences of fuzzy numbers. The main purposeof this paper is to study Tauberian theorems for the EulerN"{o}rlundmeanconvergent sequences of fuzzy numbers and investigate some other kind ofconvergences named EulerN"{o}rlund meanlevel convergence so as to fill upthe existing gaps in the literature. The results which we obtained in thisstudy are much more general than those obtained by others.
1

79
92


Naim L.
Braha
Department of Mathematics and Computer Sciences, University of
Prishtina, Avenue Mother Teresa, No4, Prishtine, 10000, Kosova
Department of Mathematics and Computer Sciences,
Serbia


Mikail
Et
Department of Mathematics, Frat University, Elazig, 23119, Turkey
Department of Mathematics, Frat University,
Turkey
mikailet68@gmail.com;mikailet@yahoo.com
Statistical convergence
Tauberian theorems
Fuzzy numbers
[[1] Y. Altin, M. Mursaleen and H. Altinok, Statistical summability (C; 1) for sequences of fuzzy##real numbers and a Tauberian theorem, Journal of Intelligent and Fuzzy Systems, 21 (2010),##[2] S. Aytar, M. A. Mammadov and S. Pehlivan, Statistical limit inferior and limit superior for##sequences of fuzzy numbers, Fuzzy Sets and Systems, 157(7) (2006), 976{985.##[3] B. Bede and S. G. Gal, Almost periodic fuzzy number valued functions, Fuzzy Sets and##Systems, 147 (2004), 385{403.##[4] N. L. Braha, Tauberian conditions under which statistical convergence follows from statistical##summability (V; ), Miskolc Math. Notes, 16(2) (2015), 695{703.##[5] M. Et, H. Altinok and R. Colak, On statistical convergence of dierence sequences of fuzzy##numbers, Inform. Sci., 176(15) (2006), 2268{2278.##[6] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241{244.##[7] J. X. Fang and H. Huang, On the level convergence of a sequence of fuzzy numbers, Fuzzy##Sets and Systems, 147(3) (2004), 417{435.##[8] A. Gokhan, M. Et and M. Mursaleen, Almost lacunary statistical and strongly almost lacunary##convergence of sequences of fuzzy numbers, Math. Comput. Modelling, 49(34) (2009),##[9] J. S. Kwon, On statistical and pCesaro convergence of fuzzy numbers, The Korean Journal##of Computational and Applied Mathematics, 7 (2000), 195{203.##[10] L. Leindler, Uber die de la ValleePousinsche summierbarkeit allgemeiner orthogonalreihen,##Acta Math. Acad. Sci. Hungar., 16 (1965), 375{387.##[11] M. Matloka, Sequence of fuzzy numbers, BUSEFAL, 28 (1986), 28{37.##[12] F. Moricz, Tauberian conditions under which statistical convergence follows from statistical##summability (C, 1), J. Math. Anal. Appl., 275 (2002), 277{287.##[13] S. Nanda, On sequence of fuzzy numbers, Fuzzy Sets and Systems, 33 (1989), 123{126.##[14] I. J. Schoenberg, The integrability of certain functions and related summability methods,##Amer. Math. Monthly, 66 (1959), 361{375.##92 N. L. Braha and M. Et##[15] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2##(1951), 73{74.##[16] C. Wu and G. Wang, Convergence of sequences of fuzzy numbers and xed point theorems for##increasing fuzzy mappings and application, Theme: Fuzzy intervals. Fuzzy Sets and Systems,##130(3) (2002), 383{390.##[17] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338{353.##[18] A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, London and##New York, 1979.##]
ON THE SYSTEM OF LEVELELEMENTS INDUCED BY AN LSUBSET
2
2
This paper focuses on the relationship between an $L$subset and the system of levelelements induced by it, where the underlying lattice $L$ is a complete residuated lattice and the domain set of $L$subset is an $L$partially ordered set $(X,P)$. Firstly, we obtain the sufficient and necessary condition that an $L$subset is represented by its system of levelelements. Then, a new representation theorem of intersectionpreserving $L$subsets is shown by using unionpreserving system of elements. At last, another representation theorem of compatible intersectionpreserving $L$subsets is obtained by means of compatible unionpreserving system of elements.
1

93
105


Jinming
Fang
Department of Mathematics, Ocean University of China, Qing Dao
266071, PR China
Department of Mathematics, Ocean University
China
jiningfang@163.com


Youyan
Li
Department of Mathematics, Ocean University of China, Qing Dao
266071, PR China
Department of Mathematics, Ocean University
China


Wenyi
Chen
Department of Mathematics, Ocean University of China, Qing Dao
266071, PR China
Department of Mathematics, Ocean University
China
ouccwy@126.com
Complete residuated lattice
$L$partially ordered set
$L$subset
System of levelelements
Unionpreserving system of elements
Compatible unionpreserving system of elements
Representation theorem
[[1] R. Belohlavek, Fuzzy relational systems: foundation and principles, New York: Kluwer Aca##demic Plenum Publishers, (2002), 75{176.##[2] J. M. Fang and Y. L. Han, A new representation theorem of Lsets, Perodical of Ocean##University of China(Natural Science), 38(6) (2008), 1025{1028.##[3] J. M. Fang, Relationships between Lordered convergence structures and strong Ltopologies,##Fuzzy Sets and Systems, 161 (2010), 2923{2944.##[4] J. M. Fang, Stratied Lordered convergence structures, Fuzzy Sets and Systems, 161 (2010),##[5] J. A. Goguen, Lfuzzy sets, J. Math. Appl., 18 (1967), 145{174.##[6] H. Han and J. M. Fang, Representation theorems of Lsubsets and Lfamilies on complete##residuated lattice, Iranian Journal of Fuzzy Systems, 10(3) (2013), 125{136.##[7] H. Lai and D. Zhang, Fuzzy preorder and fuzzy topology, Fuzzy Sets and Systems, 157 (2006),##1865{1885.##[8] C. Z. Luo, Fuzzy sets and nested systems, Journal of Fuzzy Mathematics, 3(4) (1983), 113{##[9] L. X. Meng and X. Z. Wang, An improved representation theorem of Lfuzzy set, Fuzzy Sets##and Systems, 161 (2010), 3134{3147.##[10] F. G. Shi, Theory of Lnested sets and Lnested sets and applications, Fuzzy Systems##and Mathematics, 9(4) (1995), 65{72.##[11] B. Seselja and A. Tepavcevic, A note on a natural equivalence relation on fuzzy power set,##Fuzzy Sets and Systems, 148(2) (2004), 201{210.##[12] B. Seselja and A. Tepavcevic, Representing ordered structures by fuzzy sets: an overview,##Fuzzy Sets and Systems, 136(1) (2003), 21{39.##[13] B. Seselja and A. Tepavcevc, Completion of ordered structures by cuts of fuzzy sets: an##overview, Fuzzy Sets and Systems, 136(1) (2003), 1{19.##[14] F. L. Xiong, The representation theorems on complete lattice and their application, Perodical##of Ocean University of Qingdao, 28(2) (1998), 339{344.##[15] W. Yao, Quantitative domain via fuzzy sets: part I: continuity of fuzzy completed directed##posets, Fuzzy Sets and Systems, 161 (2010), 973{987.##[16] W. Y. Zeng and Y. Shi, A kind of approach to new representation theorem, Journal of Beijing##Normal University (Natural Science), 39(1) (2003), 34{39.##[17] D. Zhang, An enriched category approach to many valued topology, Fuzzy Sets and Systems,##158 (2007), 349{366.##]
FUZZY FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS IN PARTIALLY ORDERED METRIC SPACES
2
2
In this paper, we consider fuzzy fractional partial differential equations under Caputo generalized Hukuhara differentiability. Some new results on the existence and uniqueness of two types of fuzzy solutions are studied via weakly contractive mapping in the partially ordered metric space. Some application examples are presented to illustrate our main results.
1

107
126


Hoang Viet
Long
Division of Computational Mathematics and Engineering, Insti
tute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam; Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh
City, Vietnam
Division of Computational Mathematics and
Viet Nam


Nguyen Thi Kim
Son
Department of Mathematics, Hanoi University of Education,
Vietnam
Department of Mathematics, Hanoi University
Viet Nam


Ngo Van
Hoa
Division of Computational Mathematics and Engineering, Institute
for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam; Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
Division of Computational Mathematics and
Viet Nam
Fractional PDEs
Caputo gHderivatives
Fuzzy weak solutions
Weakly contractive mapping
Partially ordered space
[[1] S. Abbas, M. Benchohra and G. M. N'Guerekata, Topics in fractional DEs, Springer, Berlin,##Heidelberg, New York, Hong Kong, London, Milan, Paris, Tokyo, 2012.##[2] R. Alikhani and F. Bahrami, Global solutions of fuzzy integrodierential equations under##generalized dierentiability by the method of upper and lower solutions, Inf. Sci., 295 (2015),##[3] T. Allahviranloo, Z. Gouyandeh and A. Armand, Fuzzy fractional dierential equations under##generalized fuzzy Caputo derivative, J. Intell. Fuzzy Syst., 26 (2014), 14811490.##[4] T. Allahviranloo, Z. Gouyandeh, A. Armand and A. Hasanoglu, On fuzzy solutions for heat##equation based on generalized Hukuhara dierentiability, Fuzzy Sets Syst., 265 (2015), 123.##[5] B. Bede and L. Stefanini, Generalized dierentiability of fuzzyvalued functions, Fuzzy Sets##Syst., 230 (2013), 119141. ##[6] M. Caputo, Linear models of dissipation whose Q is almost frequency independentII, Geo##physical J. Int., 13 (1967), 529539.##[7] J. Harjani and K. Sadarangani, Generalized contractions in partially ordered metric spaces##and applications to ordinary dierential equations, Nonlinear Anal. (TMA), 72 (2010), 1188##[8] N. V. Hoa, Fuzzy fractional functional dierential equations under Caputo gH##dierentiability, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 11341157.##[9] N. V. Hoa, Fuzzy fractional functional integral and dierential equations, Fuzzy Sets Syst.,##280 (2015), 5890.##[10] A. Khastan, J. J. Nieto and R. RodrguezLopez, Schauder xedpoint theorem in semilinear##spaces and its application to fractional dierential equations with uncertainty, Fixed Point##Theory Appl., 2014 (2014): 21.##[11] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional##dierential equations, Elsevier Science B.V, Amsterdam, 2006.##[12] H. V. Long, N. T. K. Son, N. T. M. Ha and L. H. Son, The existence and uniqueness of fuzzy##solutions for hyperbolic partial dierential equations, Fuzzy Optim. Decis. Mak., 13 (2014),##[13] H. V. Long, N. T. K. Son and H. T. T. Tam, Global existence of solutions to fuzzy partial##hyperbolic functional dierential equations with generalized Hukuhara derivatives, J. Intell.##Fuzzy Syst., 29 (2015), 939954.##[14] H. V. Long, N. T. K. Son and H. T. T. Tam, The solvability of fuzzy fractional partial##dierential equations under Caputo gHdierentiability, Fuzzy Sets Syst., 309 (2017), 3563.##[15] V. Lupulescu, Fractional calculus for intervalvalued functions, Fuzzy Sets Syst., 265 (2015),##[16] M. T. Malinowski, Random fuzzy fractional integral equations  Theoretical foundations,##Fuzzy Sets Syst., 265 (2015) 3962.##[17] J. J. Nieto and R. RodrguezLopez, Applications of contractivelike mapping principles to##fuzzy equations, Revista Matematica Complutense, 19 (2006), 361383.##[18] E. J. VillamizarRoa, V. AnguloCastillo and Y. ChalcoCano, Existence of solutions to fuzzy##dierential equations with generalized Hukuhara derivative via contractivelike mapping prin##ciples, Fuzzy Sets Syst., 265 (2015), 2438.##[19] H. Vu and N. V. Hoa, On impulsive fuzzy functional dierential equations, Iranian Journal##of Fuzzy Systems, 13(4) (2016), 7994.##]
SAPPROXIMATION SPACES: A FUZZY APPROACH
2
2
In this paper, we study the concept of Sapproximation spaces in fuzzy set theory and investigate its properties. Along introducing three pairs of lower and upper approximation operators for fuzzy Sapproximation spaces, their properties under different assumptions, e.g. monotonicity and weak complement compatibility are studied. By employing two thresholds for minimum acceptance accuracy and maximum rejection error, these spaces are interpreted in threeway decision systems by defining the corresponding positive, negative and boundary regions.
1

127
154


Ali
Shakiba
Department of Computer Science, ValieAsr University of Rafsanjan,
Rafsanjan, Iran
Department of Computer Science, ValieAsr
Iran
ali.shakiba@vru.ac.ir


MohammadReza
Hooshmandasl
Department of Computer Science, Yazd University,
Yazd, Iran
Department of Computer Science, Yazd University,
Y
Iran


Bijan
Davvaz
Department of Mathematics, Yazd University, Yazd, Iran
Department of Mathematics, Yazd University,
Iran
davvaz@yahoo.com


Seyed Abolfazl
Shahzadeh Fazeli
Department of Computer Science, Yazd University, Yazd, Iran
Department of Computer Science, Yazd University,
Iran
Fuzzy Sapproximation Spaces
Fuzzy sets
Threeway Decisions
Monotonicity
Weak Complement Compatibility
Rough Set Theory
Rough Mereology
[[1] M. Alamuri, B. R. Surampudi and A. Negi, A survey of distance/similarity measures for##categorical data, In 2014 International Joint Conference on Neural Networks, IJCNN (2014),##19071914.##[2] N. Azam and J. T. Yao, Analyzing uncertainties of probabilistic rough set regions with game##theoretic rough sets, International Journal of Approximate Reasoning, 55(1) (2014), 142155,##[3] C. Cornelis, M. De Cock and A. M. Radzikowska, Fuzzy rough sets: from theory into practice,##Handbook of Granular computing, (2008), 533552.##[4] B. Davvaz, A short note on algebraic Trough sets, Information Sciences, 178 (2008), 3247##[5] B. Davvaz, Approximations in nary algebraic systems, Soft Computing, 12(4) (2008), 409##[6] B. Davvaz, Approximations in a semigroup by using a neighbourhood system, International##Journal of Computer Mathematics, 88(4) (2011), 709713.##[7] B. Davvaz and M. Mahdavipour, Rough approximations in a general approximation space##and their fundamental properties, International Journal of General Systems, 37(3) (2008),##[8] A. P. Dempster, Upper and lower probabilities induced by a multivalued mapping, The Annals##of Mathematical Statistics, 38(2) (1967), 325{339.##[9] D. Dubois and H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of##General Systems, 17(23) (1990), 191{209.##[10] A. Gomolinska, Approximation spaces based on relations of similarity and dissimilarity of##objects, Fundamenta Informaticae, 79(3) (2007), 319{333.##[11] A. Gomolinska, On certain rough inclusion functions, Transactions on Rough Sets IX,##Springer Berlin Heidelberg, (2008), 35{55.##[12] A. Gomolinska, Rough approximation based on weak qRIFs, Transactions on Rough Sets X,##Springer Berlin Heidelberg, (2009), 117{135.##[13] A. Gomolinska and M.Wolski, Rough inclusion functions and similarity indices, Fundamenta##Informaticae, 133(2) (2014), 149{163.##[14] S. Greco, B. Matarazzo and R. Slowinski, Rough approximation of a preference relation by##dominance relations, European Journal of Operational Research, 117(1) (1999), 63{83.##[15] J. W. Grzymala Busse, Knowledge acquisition under uncertainty { a rough set approach,##Journal of Intelligent and Robotic Systems, 1(1) (1988), 3{16.##[16] J. W. Grzymala Busse, Roughset and DempsterShafer approaches to knowledge acquisition##under uncertainty { a comparison, Technical report, University of Kansas, 1987.##[17] M. R. Hooshmandasl, A. Shakiba, A. K. Goharshady and A. Karimi, Sapproximation: A##new approach to algebraic approximation, Journal of Discrete Mathematics, 2014 (2014),##[18] M. J. Lesot, M. Rifqi and H. Benhadda, Similarity measures for binary and numerical data:##a survey, International Journal of Knowledge Engineering and Soft Data Paradigms, 1(1)##(2009), 63{84.##[19] T. J. Li, Rough approximation operators on two universes of discourse and their fuzzy ex##tensions, Fuzzy Sets and Systems, 159(22) (2008), 3033{3050.##[20] P. Lingras and R. Jensen, Survey of rough and fuzzy hybridization, IEEE International Fuzzy##Systems Conference, (2007), 1{6.##[21] C. Liu, D. Miao and N. Zhang, Graded rough set model based on two universes and its##properties, KnowledgeBased Systems, 33(0) (2012), 65 { 72.##[22] G. Liu, Rough set theory based on two universal sets and its applications, KnowledgeBased##Systems, 23(2) (2010), 110{1150.##[23] N. N. Morsi and M. M. Yakout, Axiomatics for fuzzy rough sets, Fuzzy Sets and Systems,##100(1) (1998), 327{342.##[24] A. Nakamura, Fuzzy rough sets, Note on Multiplevalued Logic in Japan, 9(8) (1988), 1{8. ##[25] S. Nanda and S. Majumdar, Fuzzy rough sets, Fuzzy Sets and Systems, 45(2) (1992), 157{##[26] S. K. Pal, L. Polkowski and A. Skowron, RoughNeural Computing: Techniques for Comput##ing with Words, Articial intelligence, Springer Berlin Heidelberg, 2004.##[27] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning About Data, Kluwer Academic##Publishers, 1991.##[28] Z. Pawlak and A. Skowron, Rough membership functions, Advances in the DempsterShafer##Theory of Evidence, John Wiley & Sons, (1994), 251{271.##[29] Z. Pawlak, S. K. M. Wong, and W. Ziarko, Rough sets: Probabilistic versus deterministic##approach, International Journal of ManMachine Studies, 29 (1988), 81{95.##[30] Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences, 11##(1982), 341{356.##[31] Z. Pawlak, Vagueness and uncertainty: A rough set perspective, Computational Intelligence,##11(2) (1995), 227{232.##[32] Z. Pawlak, Rough set theory and its applications to data analysis, Cybernetics & Systems,##29(7) (1998), 661{688.##[33] Z. Pei and Z. B. Xu, Rough set models on two universes, International Journal of General##Systems, 33(5) (2004), 569{581.##[34] L. Polkowski, Rough sets: Mathematical foundations, Physica Verlag, 15 (2002).##[35] L. Polkowski and A. Skowron, Rough mereology: A new paradigm for approximate reasoning,##International Journal of Approximate Reasoning, 15(4) (1996), 333{365.##[36] L. Polkowski and A. Skowron, Rough mereology, Methodologies for Intelligent Systems,##Springer Berlin Heidelberg, (1994), 85{94.##[37] A. M. Radzikowska and E. E. Kerre, A comparative study of fuzzy rough sets, Fuzzy Sets and##Systems, 126(2) (2002), 137{155.##[38] A. M. Radzikowska and E. E. Kerre, A comparative study of fuzzy rough sets, Fuzzy Sets and##Systems, 126(2) (2002), 137{155.##[39] G. Shafer, A mathematical theory of evidence, Princeton university press Princeton, 1 (1976).##[40] A. Shakiba and M. R. Hooshmandasl, Sapproximation spaces: A threeway decision ap##proach, Fundamenta Informaticae, 139(3) (2015), 307{328.##[41] A. Shakiba and M. R. Hooshmandasl, Neighborhood system Sapproximation spaces and##applications, Knowledge and Information Systems, (2015), 1{46.##[42] Y. Shen and F. Wang, Variable precision rough set model over two universes and its proper##ties, Soft Computing, 15(3) (2011),557{567.##[43] A. Skowron and J. Stepaniuk, Tolerance approximation spaces, Fundamenta Informaticae,##27(2) (1996), 245{253.##[44] R. Slowinski and D. Vanderpooten, A generalized denition of rough approximations based on##similarity, IEEE Transactions on Knowledge and Data Engineering, 12(2) (2000), 331{336.##[45] B. Sun, Z. Gong and D. Chen, Fuzzy rough set theory for the intervalvalued fuzzy information##systems, Information Sciences, 178(13) (2008), 2794{2815.##[46] B. Sun and W. Ma, Fuzzy rough set model on two dierent universes and its application,##Applied Mathematical Modelling, 35(4) (2011), 1798{1809.##[47] S. K. M. Wong, L. S. Wang, and Y. Y. Yao, Interval structure: A framework for representing##uncertain information, Proceedings of the Eighth International Conference on Uncertainty##in Articial Intelligence, UAI'92 (1992), 336{343.##[48] W. Z.Wu, J. Sheng Mi and W. Xiu Zhang, Generalized fuzzy rough sets, Information Sciences,##151 (2003), 263{282.##[49] W. Xu, W. Sun, Y. Liu and W. Zhang, Fuzzy rough set models over two universes, Interna##tional Journal of Machine Learning and Cybernetics, 4(6) (2013), 631{645.##[50] S. Yamak, O. Kazanci and B. Davvaz, Generalized lower and upper approximations in a ring,##Information Sciences, 180(9) (2010), 1759{1768.##[51] R. Yan, J. Zheng, J. Liu and C. Qin, Rough set over dualuniverses in fuzzy approximation##space, Iranian Journal of Fuzzy Systems, 9(3) (2012), 79{91 ##[52] Q. Yang and X. Wu, 10 challenging problems in data mining research, International Journal##of Information Technology & Decision Making, 5(4) (2006), 597{604.##[53] Y. Y. Yao, A comparative study of fuzzy sets and rough sets, Information Sciences, 109(14)##(1998), 227{242.##[54] Y. Y. Yao, Probabilistic approaches to rough sets, Expert Systems, 20(5) (2003), 287{297.##[55] Y. Y. Yao and P. J. Lingras, Interpretations of belief functions in the theory of rough sets,##Information Sciences, 104(12) (1998), 81{106.##[56] Y. Y. Yao, Probabilistic rough set approximations, International Journal of Approximate##Reasoning, 49(2) (2008), 255{271.##[57] Y. Y. Yao, Threeway decision: an interpretation of rules in rough set theory, International##Conference on Rough Sets and Knowledge Technology, Springer Berlin Heidelberg, (2009),##[58] Y. Y. Yao, An outline of a theory of threeway decisions, Rough Sets and Current Trends in##Computing, Springer Berlin Heidelberg, (2012), 1{17.##[59] Y. Y. Yao and X. Deng, Quantitative rough sets based on subsethood measures, Information##Sciences, 267 (2014), 306{322.##[60] Y. Y. Yao, Combination of rough and fuzzy sets based on level sets, Rough sets and Data##Mining, Springer, (1997), 301{321.##[61] Y. Y. Yao, Generalized rough set models, Rough Sets in Knowledge Discovery 1: Methodology##and Approximations, Studies in Fuzziness and Soft Computing, Physica Verlag Heidelberg,##1 (1998), 286{318.##[62] Y. Y. Yao and T. Y. Lin, Generalization of rough sets using modal logic, Intelligent Automa##tion and Soft Computing, 2(2) (1996), 103{120.##[63] L. A. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338{353.##[64] L. A. Zadeh, The role of fuzzy logic in the management of uncertainty in expert systems,##Fuzzy Sets and Systems, 11(1) (1983), 197{198.##[65] L. A. Zadeh, A simple view of the DempsterShafer theory of evidence and its implication##for the rule of combination, AI magazine, 7(2) (1986), 85{90.##[66] H. Y. Zhang, W. X. Zhang and W. Z. Wu, On characterization of generalized interval##valued fuzzy rough sets on two universes of discourse, International Journal of Approximate##Reasoning, 51(1) (2009), 56{70.##[67] W. X. Zhang and Y. Leung, Theory of including degrees and its applications to uncertainty##inferences, Fuzzy Systems Symposium, Soft Computing in Intelligent Systems and Informa##tion Processing., Proceedings of the 1996 Asian, (1996), 496{501.##[68] W. Zhu and F. Y.Wang, A new type of covering rough set, 3rd International IEEE Conference##on Intelligent Systems, (2006), 444{449.##[69] W. Zhu and F. Y. Wang, On three types of coveringbased rough sets, IEEE Transactions on##Knowledge and Data Engineering, 19(8) (2007), 1131{1144.##[70] W. Zhu and F. Y. Wang, The fourth type of coveringbased rough sets, Information Sciences,##201 (2012), 80{92.##[71] W. Ziarko, Variable precision rough set model, Journal of Computer and System Sciences,##46(1) (1993), 39{59.##[72] W. Ziarko, Probabilistic decision tables in the variable precision rough set model, Computa##tional Intelligence, 17(3) (2001), 593{603.##]
FORMAL BALLS IN FUZZY PARTIAL METRIC SPACES
2
2
In this paper, the poset $BX$ of formal balls is studied in fuzzy partial metric space $(X,p,*)$. We introduce the notion of layered complete fuzzy partial metric space and get that the poset $BX$ of formal balls is a dcpo if and only if $(X,p,*)$ is layered complete fuzzy partial metric space.
1

155
164


Jiyu
Wu
Department of Mathematics, Ocean University of China, 238 Songling
Road, 266100, Qingdao, P.R.China
Department of Mathematics, Ocean University
China
wjytun@aliyun.com


Yueli
Yue
Department of Mathematics, Ocean University of China, 238 Songling
Road, 266100, Qingdao, P.R.China
Department of Mathematics, Ocean University
China
ylyue@ouc.edu.cn
Fuzzy partial metric
Formal ball
$mathcal{Q}$category
Domain
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