IJFS University of Sistan and Baluchestan Iranian Journal of Fuzzy Systems 1735-0654 University of Sistan and Baluchestan 21 10.22111/ijfs.2017.3139 unavailable Cover Vol.14, No.2 April 2017 29 04 2017 14 2 0 0 23 04 2017 23 04 2017 Copyright © 2017, University of Sistan and Baluchestan. 2017 http://ijfs.usb.ac.ir/article_3139.html

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IJFS University of Sistan and Baluchestan Iranian Journal of Fuzzy Systems 1735-0654 University of Sistan and Baluchestan 21 10.22111/ijfs.2017.3131 Research Paper SOME RESULTS OF MOMENTS OF UNCERTAIN RANDOM VARIABLES Ahmadzade Hamed Department of Statistics, University of Sistan and Baluchestan, Zahedan, Iran Sheng Yuhong College of Mathematical and System Sciences, Xinjiang University, Urumqi 830046, China Hassantabar Darzi Fatemeh Department of Statistics, University of Sistan and Baluchestan, Zahedan, Iran 29 04 2017 14 2 1 21 22 10 2015 22 10 2016 Copyright © 2017, University of Sistan and Baluchestan. 2017 http://ijfs.usb.ac.ir/article_3131.html

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.

Chance theory Uncertain random variable Chance distribution Moments
 X. Chen and W. Dai, Maximum entropy principlefor uncertain variables, International Jour- nal of Fuzzy Systems, 13(3) (2011), 232{236.  X. Chen, S. Kar and D. Ralescu, Cross-entropy measure of uncertain variables, Information Sciences, 201 (2012), 53{60.  X. Chen and D. Ralescu, Liu process and uncertain calculus, Journal of Uncertainty Analysis and Applications, 1(3) (2013), 1{ 12.  W. Dai and X. Chen, Entropy of function of uncertain variables, Mathematics and Computer Modelling, 55 (2012), 754{760.  H. Y. Guo and X. S. Wang, Variance of uncertain random variables, Journal of Uncertainty Analysis and Applications, 2(6) (2014), 1{7.  Y. C. Hou, Subadditivity of chance measur, Journal of Uncertainty Analysis and Applications, 2(14) (2014), 1{8.  A. N. Kolmogorov, Grundbegri e der Wahrscheinlichkeitsrechnung, Julius Springer, Berlin,  R. Kruse and K. Meyer, Statistics with Vague Data, Reidel Publishing Company, Dordrecht,  B. Liu, Uncertainty Theory, 5th ed., http://orsc.edu.cn/liu/ut.pdf. 2014.  B. Liu, Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, 2007.  B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 31 (2009), 3{10.  B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010.  B. Liu, Toward uncertain nance theory, Journal of Uncertainty Analysis and Applications, 1(1) (2013), 1{15.  Y. H. Liu, Uncertain random variables: A mixture of uncertainty and randomness, Soft Computing, 17(4) (2013), 625{634.  Y. H. Liu, Uncertain random programming with applications, Fuzzy Optimization and Deci- sion Making, 12(2) (2013), 153{169.  Y. H. Liu and M. H. Ha, Expected value of function of uncertain variables, Journal of Un- certain Systems, 4(3) (2010), 181{186.  Y. K. Liu and B. Liu, Fuzzy random variables: a scalar expected value operator, Fuzzy Optimization and Decision Making, 2(2) (2003), 143{160.  Y. K. Liu and B. Liu, Fuzzy random programming with equilibrium chance constraints, In- formtion Sciences 170 (2005), 363{395.  Z. X. Peng and K. Iwamura, A sucient and necessary condition of uncertainty distribution, Journal of Interdisciplinary Mathematics, 13(3) (2010), 277{285.  M. Puri and D. Ralescu, Fuzzy random variables, Journal of Mathmatical Application, 114 (1986), 409{422.  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.  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.  J. L. Teugels and B. Sundt, Encyclopedia of actuarial science, Wiley & Sons, 1 (2004).  M. Wen and R. Kang, Reliability analysis in uncertain random system, Fuzzy Optimization and Decision Making, doi:10.1007/s10700-016-9235-y, (2016).  K. Yao, A formula to calculate the variance of uncertain variable, Soft Computing, 19(10) (2015), 2947{2953.
IJFS University of Sistan and Baluchestan Iranian Journal of Fuzzy Systems 1735-0654 University of Sistan and Baluchestan 21 10.22111/ijfs.2017.3132 Research Paper A JOINT DUTY CYCLE SCHEDULING AND ENERGY AWARE ROUTING APPROACH BASED ON EVOLUTIONARY GAME FOR WIRELESS SENSOR NETWORKS Kordafshari M. S. Department of Computer Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran Movaghar A. Department of Computer Engineering, Sharif University of Technology, Tehran, Iran Meybodi M. R. Computer Engineering and Information Technology Department, Amirkabir University of Technology, Tehran, Iran 29 04 2017 14 2 23 44 22 04 2016 22 10 2016 Copyright © 2017, University of Sistan and Baluchestan. 2017 http://ijfs.usb.ac.ir/article_3132.html

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 trade-off  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 trade-off 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.

Wireless sensor network Duty cycle scheduling Energy aware routing Evolutionary game theory Distributed reinforcement learning
IJFS University of Sistan and Baluchestan Iranian Journal of Fuzzy Systems 1735-0654 University of Sistan and Baluchestan 21 10.22111/ijfs.2017.3133 Research Paper MULTI-OBJECTIVE ROUTING AND SCHEDULING IN FLEXIBLE MANUFACTURING SYSTEMS UNDER UNCERTAINTY Mehrabian Ahmad Department of Industrial Engineering, South-Tehran Branch, Islamic Azad University, Tehran, Iran Tavakkoli-Moghaddam Reza Department of Industrial Engineering, South-Tehran Branch, Islamic Azad University, Tehran, Iran Khalili-Damaghani Kaveh Department of Industrial Engineering, South-Tehran Branch, Islamic Azad University, Tehran, Iran 29 04 2017 14 2 45 77 22 04 2016 22 10 2016 Copyright © 2017, University of Sistan and Baluchestan. 2017 http://ijfs.usb.ac.ir/article_3133.html

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 two-objective mathematical programming model is presented to integrate flow shop scheduling and routing AVGs in a flexible manufacturing system. In real-life 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 meta-heuristic algorithms of Non-dominated Sorting Genetic Algorithm-II (NSGAII) and multi-objective particle swarm optimization (MOPSO) are offered that the accuracy of mathematical models and efficiency of algorithms provided are assessed through numerical examples.

Scheduling Routing Automated guided vehicle meta-heuristic algorithm Flexible manufacturing
IJFS University of Sistan and Baluchestan Iranian Journal of Fuzzy Systems 1735-0654 University of Sistan and Baluchestan 21 10.22111/ijfs.2017.3134 Research Paper TAUBERIAN THEOREMS FOR THE EULER-NORLUND MEAN-CONVERGENT SEQUENCES OF FUZZY NUMBERS Braha Naim L. Department of Mathematics and Computer Sciences, University of Prishtina, Avenue Mother Teresa, No-4, Prishtine, 10000, Kosova Et Mikail Department of Mathematics, Frat University, Elazig, 23119, Turkey 29 04 2017 14 2 79 92 22 03 2016 22 10 2016 Copyright © 2017, University of Sistan and Baluchestan. 2017 http://ijfs.usb.ac.ir/article_3134.html

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 Euler-N"{o}rlund mean-convergent 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 Euler-N"{o}rlundmean-convergent sequences of fuzzy numbers and investigate some other kind ofconvergences named Euler-N"{o}rlund mean-level 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.

Statistical convergence Tauberian theorems Fuzzy numbers
 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),  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.  B. Bede and S. G. Gal, Almost periodic fuzzy number valued functions, Fuzzy Sets and Systems, 147 (2004), 385{403.  N. L. Braha, Tauberian conditions under which 􀀀statistical convergence follows from statistical summability (V; ), Miskolc Math. Notes, 16(2) (2015), 695{703.  M. Et, H. Altinok and R. Colak, On -statistical convergence of di erence sequences of fuzzy numbers, Inform. Sci., 176(15) (2006), 2268{2278.  H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241{244.  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.  A. Gokhan, M. Et and M. Mursaleen, Almost lacunary statistical and strongly almost lacunary convergence of sequences of fuzzy numbers, Math. Comput. Modelling, 49(3-4) (2009),  J. S. Kwon, On statistical and p-Cesaro convergence of fuzzy numbers, The Korean Journal of Computational and Applied Mathematics, 7 (2000), 195{203.  L. Leindler,  Uber die de la Vallee-Pousinsche summierbarkeit allgemeiner orthogonalreihen, Acta Math. Acad. Sci. Hungar., 16 (1965), 375{387.  M. Matloka, Sequence of fuzzy numbers, BUSEFAL, 28 (1986), 28{37.  F. Moricz, Tauberian conditions under which statistical convergence follows from statistical summability (C, 1), J. Math. Anal. Appl., 275 (2002), 277{287.  S. Nanda, On sequence of fuzzy numbers, Fuzzy Sets and Systems, 33 (1989), 123{126.  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  H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951), 73{74.  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.  L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338{353.  A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, London and New York, 1979.
IJFS University of Sistan and Baluchestan Iranian Journal of Fuzzy Systems 1735-0654 University of Sistan and Baluchestan 21 10.22111/ijfs.2017.3135 Research Paper ON THE SYSTEM OF LEVEL-ELEMENTS INDUCED BY AN L-SUBSET Fang Jinming Department of Mathematics, Ocean University of China, Qing Dao 266071, PR China Li Youyan Department of Mathematics, Ocean University of China, Qing Dao 266071, PR China Chen Wenyi Department of Mathematics, Ocean University of China, Qing Dao 266071, PR China 29 04 2017 14 2 93 105 22 10 2014 22 10 2016 Copyright © 2017, University of Sistan and Baluchestan. 2017 http://ijfs.usb.ac.ir/article_3135.html

This paper focuses on the relationship between an \$L\$-subset and the system of level-elements 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 level-elements. Then, a new representation theorem of intersection-preserving \$L\$-subsets is shown by using union-preserving system of elements. At last, another representation theorem of compatible intersection-preserving \$L\$-subsets is obtained by means of compatible union-preserving system of elements.

Complete residuated lattice \$L\$-partially ordered set \$L\$-subset System of level-elements Union-preserving system of elements Compatible union-preserving system of elements Representation theorem
 R. Belohlavek, Fuzzy relational systems: foundation and principles, New York: Kluwer Aca- demic Plenum Publishers, (2002), 75{176.  J. M. Fang and Y. L. Han, A new representation theorem of L-sets, Perodical of Ocean University of China(Natural Science), 38(6) (2008), 1025{1028.  J. M. Fang, Relationships between L-ordered convergence structures and strong L-topologies, Fuzzy Sets and Systems, 161 (2010), 2923{2944.  J. M. Fang, Strati ed L-ordered convergence structures, Fuzzy Sets and Systems, 161 (2010),  J. A. Goguen, L-fuzzy sets, J. Math. Appl., 18 (1967), 145{174.  H. Han and J. M. Fang, Representation theorems of L-subsets and L-families on complete residuated lattice, Iranian Journal of Fuzzy Systems, 10(3) (2013), 125{136.  H. Lai and D. Zhang, Fuzzy preorder and fuzzy topology, Fuzzy Sets and Systems, 157 (2006), 1865{1885.  C. Z. Luo, Fuzzy sets and nested systems, Journal of Fuzzy Mathematics, 3(4) (1983), 113{  L. X. Meng and X. Z. Wang, An improved representation theorem of L-fuzzy set, Fuzzy Sets and Systems, 161 (2010), 3134{3147.  F. G. Shi, Theory of L-nested sets and L -nested sets and applications, Fuzzy Systems and Mathematics, 9(4) (1995), 65{72.  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.  B. Seselja and A. Tepavcevic, Representing ordered structures by fuzzy sets: an overview, Fuzzy Sets and Systems, 136(1) (2003), 21{39.  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.  F. L. Xiong, The representation theorems on complete lattice and their application, Perodical of Ocean University of Qingdao, 28(2) (1998), 339{344.  W. Yao, Quantitative domain via fuzzy sets: part I: continuity of fuzzy completed directed posets, Fuzzy Sets and Systems, 161 (2010), 973{987.  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.  D. Zhang, An enriched category approach to many valued topology, Fuzzy Sets and Systems, 158 (2007), 349{366.
IJFS University of Sistan and Baluchestan Iranian Journal of Fuzzy Systems 1735-0654 University of Sistan and Baluchestan 21 10.22111/ijfs.2017.3136 Research Paper FUZZY FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS IN PARTIALLY ORDERED METRIC SPACES Long Hoang Viet 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 Son Nguyen Thi Kim Department of Mathematics, Hanoi University of Education, Vietnam Hoa Ngo Van 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 29 04 2017 14 2 107 126 22 02 2016 22 10 2016 Copyright © 2017, University of Sistan and Baluchestan. 2017 http://ijfs.usb.ac.ir/article_3136.html

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.

Fractional PDEs Caputo gH-derivatives Fuzzy weak solutions Weakly contractive mapping Partially ordered space
 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.  R. Alikhani and F. Bahrami, Global solutions of fuzzy integro-di erential equations under generalized di erentiability by the method of upper and lower solutions, Inf. Sci., 295 (2015),  T. Allahviranloo, Z. Gouyandeh and A. Armand, Fuzzy fractional di erential equations under generalized fuzzy Caputo derivative, J. Intell. Fuzzy Syst., 26 (2014), 1481-1490.  T. Allahviranloo, Z. Gouyandeh, A. Armand and A. Hasanoglu, On fuzzy solutions for heat equation based on generalized Hukuhara di erentiability, Fuzzy Sets Syst., 265 (2015), 1-23.  B. Bede and L. Stefanini, Generalized di erentiability of fuzzy-valued functions, Fuzzy Sets Syst., 230 (2013), 119-141.  M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geo- physical J. Int., 13 (1967), 529-539.  J. Harjani and K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary di erential equations, Nonlinear Anal. (TMA), 72 (2010), 1188-  N. V. Hoa, Fuzzy fractional functional di erential equations under Caputo gH- di erentiability, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 1134-1157.  N. V. Hoa, Fuzzy fractional functional integral and di erential equations, Fuzzy Sets Syst., 280 (2015), 58-90.  A. Khastan, J. J. Nieto and R. Rodrguez-Lopez, Schauder xed-point theorem in semilinear spaces and its application to fractional di erential equations with uncertainty, Fixed Point Theory Appl., 2014 (2014): 21.  A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional di erential equations, Elsevier Science B.V, Amsterdam, 2006.  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 di erential equations, Fuzzy Optim. Decis. Mak., 13 (2014),  H. V. Long, N. T. K. Son and H. T. T. Tam, Global existence of solutions to fuzzy partial hyperbolic functional di erential equations with generalized Hukuhara derivatives, J. Intell. Fuzzy Syst., 29 (2015), 939-954.  H. V. Long, N. T. K. Son and H. T. T. Tam, The solvability of fuzzy fractional partial di erential equations under Caputo gH-di erentiability, Fuzzy Sets Syst., 309 (2017), 35-63.  V. Lupulescu, Fractional calculus for interval-valued functions, Fuzzy Sets Syst., 265 (2015),  M. T. Malinowski, Random fuzzy fractional integral equations - Theoretical foundations, Fuzzy Sets Syst., 265 (2015) 39-62.  J. J. Nieto and R. Rodrguez-Lopez, Applications of contractive-like mapping principles to fuzzy equations, Revista Matematica Complutense, 19 (2006), 361-383.  E. J. Villamizar-Roa, V. Angulo-Castillo and Y. Chalco-Cano, Existence of solutions to fuzzy di erential equations with generalized Hukuhara derivative via contractive-like mapping prin- ciples, Fuzzy Sets Syst., 265 (2015), 24-38.  H. Vu and N. V. Hoa, On impulsive fuzzy functional di erential equations, Iranian Journal of Fuzzy Systems, 13(4) (2016), 79-94.
IJFS University of Sistan and Baluchestan Iranian Journal of Fuzzy Systems 1735-0654 University of Sistan and Baluchestan 21 10.22111/ijfs.2017.3137 Research Paper S-APPROXIMATION SPACES: A FUZZY APPROACH Shakiba Ali Department of Computer Science, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran Hooshmandasl MohammadReza Department of Computer Science, Yazd University, Yazd, Iran Davvaz Bijan Department of Mathematics, Yazd University, Yazd, Iran Shahzadeh Fazeli Seyed Abolfazl Department of Computer Science, Yazd University, Yazd, Iran 29 04 2017 14 2 127 154 22 06 2015 22 10 2016 Copyright © 2017, University of Sistan and Baluchestan. 2017 http://ijfs.usb.ac.ir/article_3137.html

In this paper, we study the concept of S-approximation spaces in fuzzy set theory and investigate its properties. Along introducing three pairs of lower and upper approximation operators for fuzzy S-approximation 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 three-way decision systems by defining the corresponding positive, negative and boundary regions.

Fuzzy S-approximation Spaces Fuzzy sets Three-way Decisions Monotonicity Weak Complement Compatibility Rough Set Theory Rough Mereology