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Appl. 24: 182–190\nCiungu L C (2006) Classes of residuated lattices. Annals of University of Craiova. Math. Comp. Sci. Ser. 33: 189–207\nDilworth R P (1939) Non-commutative residuated lattices. Trans. Amer. Math. Soc. 46(3): 426–444\nDymek G (2008) Fuzzy prime ideals of pseudo MV-algebras. Soft Computing 12: 365–372\nGeorgescu G, Iorgulescu A (1999) Pseudo MV-algebras: a non-commutative extension of MValgebras. In: The Proc. of the 4th International Symposium on Economic Informatics, Bucharest: 961–968\nGeorgescu G, Iorgulescu A (2001) Pseudo MV-algebras. Mult. Val. Logic (A special issue dedicated to the memory of Gr. C. Moisil) 6(1–2): 95–135\nHájek P (1998) Metamathematics of fuzzy logic. Kluwer Academic Publishers, Dordrecht\nHoo C S (1994) Fuzzy implicative and Boolean ideals of MV-algebras. Fuzzy Sets and Systems 66: 315–327\nHoo C S (1994) Fuzzy ideals of BCI and MV-algebras. Fuzzy Sets and Systems 62: 111–114\nHoo C S (1998) Some fuzzy concepts of BCI, BCK and MV-algebras. Internat. J. Approx. Reason. 18: 177–189\nHoo C S, Sessa S (1994) Fuzzy maximal ideals of BCI and MV-algebras. Inform. Sci. 80: 299–309\nHoo C S, Sessa S (1994) Implicative and Boolean ideals in MV-algebras. Math. Jpn. 39: 215–219\nImai Y, Iséki K (1966) On axiom systems of propositional calculi XIV. Proc. Japan Academy. 42: 19–22\nIséki K, Tanaka S (1976) Ideal theory of BCK-algebras. Math. Jpn. 21: 351–366\nJun Y B, Walendziak A (2006) Fuzzy ideals of pseudo MV-algebras. Int. Rev. Fuzzy Math. 1: 21–31\nLiu L, Li K (2007) Boolean filters and positive implicative filters of residuated lattices. Inform. Sci. 177: 5725–5738\nLiu L, Li K (2005) Fuzzy Boolean and positive implicative filters of BL-algebras. Fuzzy Sets and Systems 152: 333–348\nMundici D (1986) MV-algebras are categorically equivalent to bounded commutative BCK-algebras. Math. Jpn. 31: 889–894\nMunkres J R (2000) Topology. Dorling Kindersley, India\nRachunek J (2002) A non-commutaitive generalization of MV-algebras. Czech. 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Math. Anal. Appl., 24, 182, 10.1016\u002F0022-247X(68)90057-7\nÇoker D., 1997, Fuzzy Sets and Systems, 88, 81, 10.1016\u002FS0165-0114(96)00076-0\nDubois D., 2005, Fuzzy Sets and Systems, 156, 485, 10.1016\u002Fj.fss.2005.06.001\nGarcía J.G., 2005, Fuzzy Sets and Systems, 156, 445, 10.1016\u002Fj.fss.2005.05.023\nLowen R., 1989, Fuzzy Sets and Systems, 29, 171, 10.1016\u002F0165-0114(89)90191-7\nManes E.G., 1974, General Topol. Appl., 4, 341, 10.1016\u002F0016-660X(74)90011-7\nMarny T., 1979, General Topol. Appl., 10, 175, 10.1016\u002F0016-660X(79)90006-0\nMondal T.K., 1999, Indian J. Pure Appl. Math., 30, 23\nMondal T.K., 2001, Fuzzy Sets and Systems, 119, 483, 10.1016\u002FS0165-0114(98)00436-9\nSrivastava A.K., 1984, J. Math. Anal. Appl., 103, 103, 10.1016\u002F0022-247X(84)90160-4\nSrivastava A.K., 1998, Inform. Sci., 110, 195, 10.1016\u002FS0020-0255(97)10038-X\nZadeh L.A., 1975, Inform. 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For this purpose, we take a distance between the sample mean and the mean in the null hypothesis as a test statistic. An asymptotic test about the fuzzy mean is obtained by using a central limit theorem. The asymptotical distribution is ω\n                        2-distribution. The ω\n                        2-distribution is only known for special cases, thus we have considered random LR-fuzzy numbers. In the fuzzy concept, in addition to the existence of several versions of the central limit theorem, there is another practical disadvantage: The limit law is, in most cases, difficult to handle. Therefore, the central limit theorem for fuzzy random variable does not seem to be a very useful tool to make inferences on the mean of fuzzy random variable. Thus we use the bootstrap technique. Finally, by means of a simulation study, we show that the bootstrap method is a powerful tool in the statistical hypothesis testing about the mean of fuzzy random variables.",{"EN":983},"A fuzzy bootstrap test for the mean with D p,q -distance",{"VOID":985},"[\"8472562094447307832\"]",{"VOID":987},"Colubi A (2009) Statistical inference about the means of fuzzy random variables:applications to the analysis of fuzzy and real-valued data. Fuzzy Sets and Systems 160: 344–356\nEfron B (1979) Bootstrap methods: Another look at the jackknife. Ann. Statist 7: 1–26\nGonzalez Rodriguez G, Montenegro M, Colubi A, Angeles Gil M (2004) Bootstrap techniques and fuzzy random variables: Synergy in hypothesis testing with fuzzy data. Fuzzy Sets and Systems 157: 2608–2613\nKorner R (2000) An asymptotic ϕ-tests for the expectation of random fuzzy variables. J. Statist. Plann. Inference 83: 331–346\nRao R N K, Scott A J (1981) The analysis of categorical data from complex sample surveys: chisquared tests for goodness of fit and independence in two-way tables. J. Amer. Statist. Assoc.76: 221–230\nPardo L, Zografos K (2000) Goodness of fit tests with misclassified data based on ϕ-divergences. Biometrical Journal 42: 223–237\nPuri M L, Ralescu D A (1986) Fuzzy random variables. J. Math. Anal.Appl.114: 409–422\nTanaka H, Okuda T, Asai K (1965) Fuzzy information and decision in statistical model. 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Topol., 9, 77, 10.4995\u002Fagt.2008.1871\nRodabaugh S.E., 2010, Fuzzy Sets and Systems, 161, 453, 10.1016\u002Fj.fss.2009.09.010\nK.I.RosenthalQuantales and Their Applications1990Addison Wesley Longman\nSimmons H., 1982, Topology Appl., 13, 201, 10.1016\u002F0166-8641(82)90021-9\nSmith J.D.H., 1999, Algebra Stoch. Methods, 19, 9\nSolovyov S., 2006, Fuzzy Sets and Systems, 157, 459, 10.1016\u002Fj.fss.2005.06.007\nSolovyov S., 2008, Math. Stud. (Tartu), 4, 89\nSolovyov S., 2008, Fuzzy Sets and Systems, 159, 2567, 10.1016\u002Fj.fss.2008.02.010\nSolovyov S., 2010, Acta Univ. M. Belii, Ser. Math., 17, 57\nSolovyov S., 2010, Contr. Gen. Alg., 19, 187\nSolovyov S., 2010, Fuzzy Sets and Systems, 161, 2945, 10.1016\u002Fj.fss.2010.05.007\n10.2989\u002F16073606.2011.570292\nSolovyov S., 2011, Iran. J. Fuzzy Syst., 8, 1\nSolovyov S., 2012, Fuzzy Sets and Systems, 192, 176, 10.1016\u002Fj.fss.2011.07.016\nStone M.H., 1936, Trans. Am. Math. Soc., 40, 37\nStone M.H., 1937, Cas. Mat. 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almost all the realistic circumstances, such as health risk assessment and uncertainty analysis of atmospheric dispersion, it is very essential to include all the information into modelling. The parameters associated to a particular model may include different kind of variability, imprecision and uncertainty. More often, it is seen that available informations are interpreted in probabilistic sense. Probability theory is a well-established theory to measure such kind of variability. However, not all of available information, data or model parameters affected by variability, imprecision and uncertainty can be handled by traditional probability theory. Uncertainty or imprecision may occur due to incomplete information or data, measurement errors or data obtained from expert judgement or subjective interpretation of available data or information. Thus, model parameters, data may be affected by subjective uncertainty. Traditional probability theory is inappropriate to represent them. Possibility theory and fuzzy set theory is another branch of mathematics which is used as a tool to describe the parameters with insufficient or vague knowledge. In this paper, an attempt has been made to combine probability knowledge and possibility knowledge and draw the uncertainty. The paper describes an algorithm for combining probability distribution and interval-valued fuzzy number and applied to environmental risk modelling with a case study. The primary aim of this paper is to propagate the proposed method. Computer codes are prepared for the proposed method using MATLAB.",{"EN":1593},"Environmental risk modelling under probability-normal interval-valued fuzzy number",{"VOID":1595},"Saeedi M, Fakhraee H, Sadrabadi M R (2008) A fuzzy modified Gaussian air pollution dispersion model. Research Journal of Environmental Sciences 2(3): 159–165\nZadeh L A (1965) Fuzzy sets. Information Control 8: 338–353\nFerson S, Root W, Kuhn R (1999) RAMAS risk Calc: Risk assessment with uncertain numbers: Applied Biomathematics. Setauket, New York\nMahadevan S, Raghothamachar P (2000) Simulation for system reliability analysis of large structures. Computers and Structures 77: 725–734\nHoybe J A (1998) Model error propagation and date collection design: an application in water quality modeling. Water, Air and Soil Pollution 103(1–4): 101–119\nIsukapalli S S, Georgopoulos P G (1998) Stochastic response surface methods (SRSMs) for uncertainty propagation: application to environmental and biological systems. Risk Analysis 18(3): 351–363\nWang C (1999) Parametic uncertainty analysis for complex engineering systems. Ph. D Thesis. MIT\nDu X, Chen W (2001) A most probable point based method for uncertainty analysis. Journal of Design and Manufacturing Automation 4(1): 47–66\nSengupta A, Pal T K (2000) Theory and methodology: On comparing interval numbers. European Journal of Operational research 27: 28–43\nDubois D, Parde H M(1988) Possibility theory: An approach to computerized processing of uncertainty. New York: Plenum Press\nZadeh L A (1978) Fuzzy sets as a basis for theory of possibility. Fuzzy Sets and System 1(1): 3–28\nHessian W C, Strom D E, Haan C T (1996) Two-phase uncertainty analysis: An example using the universal soil loss equation. Transactions of the ASAE 39(4): 1309–1319\nTucker WT, Ferson S (2003) Probability bounds analysis in environmental risk assessment. Setauket, New York: Applied Biomathematics\nGuyonnet D, Bourgine B, Dubois D, Fargier H, Chiles J (2003) A hybrid approach for addressing uncertainty in risk assessment. Journal of Environmental Engineering 126: 68–78\nKentel E, Aral M M (2004) Probabilistic-fuzzy health risk modelling. Stochastic Environmental Research and Risk Assessment 18: 324–338\nFaybishenko B (2010) Fuzzy-probabilistic calculations of water-balance uncertainty. Stochastic Environmental Research and Risk Assessment 24: 939–952\nChutia R, Mahanta S, Datta D (2013) Uncertainty modelling of atmospheric dispersion model using fuzzy set and imprecise probability. Journal of Intelligent and Fuzzy Systems 25: 737–746\nChen S J, Chen S M (2003) Fuzzy risk analysis on similarity measures of generalized fuzzy numbers. IEEE Transactions of Fuzzy Systems 11(1): 45–56\nChen S J, Chen S M (2008) Fuzzy risk analysis based on measures of similarity between intervalvalued fuzzy numbers. Computers & Mathematics with Applications 55: 1670–1685\nChen S J (2006) A new method for handling the similarity measure problems of interval valued fuzzy numbers. Proceedings of the Second International Conference on Natural Computation and the Third International Conference on Fuzzy Systems and Knowledge Discovery, China: 25–334\nChen S M, Chen J H (2009) Fuzzy risk based on similarity measures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operators. Expert Systems with Applications 36: 6309–6317\nWei S H, Chen S M (2009) Fuzzy risk analysis based on interval-valued fuzzy numbers. Expert Systems with Applications 36: 2285–2299\nChen S J, Chen S M (2006) A new method for ranking generalized fuzzy numbers for handling fuzzy risk analysis problems. Proceedings of the Ninth Conference of Information Sciences: 1196-1199\nChen S J, Chen S M (2006) Fuzzy risk analysis based on the ranking of generalized trapezoidal fuzzy numbers. Applied Intelligence 26(1): 1–11\nZadeh L A (1975) The concept of linguistic variables and its applications to approximate reasoning-I. Information Sciences 8: 199–249\nGorzalczany B (1983) Approximate inference with interval-valued fuzzy sets-an outline. In: Proceedings of the Polish Symposium on Interval and Fuzzy Mathematics, Poland, Poznan: 89–95\nTurksen B (1986) Interval valued fuzzy sets based on normal forms. Fuzzy Sets and Systems 20: 191–210\nMendel J M, Karnik N N (2001) Operations on type-2 fuzzy sets. Fuzzy Sets and Systems 122: 327–348\nMendel J M, John R (2001) Type-2 fuzzy sets made simple. IEEE Transactions on Fuzzy Systems 10(2): 117–127\nMizumoto M, Tanaka K (1976) Some properties of fuzzy sets of type-2. Information and Control 31: 312–340\nWang G J, Li X P (1998) The applications of interval-valued fuzzy numbers and interval-distribution numbers. Fuzzy Sets and Systems 98(3): 331–335\nHamrawi H, Coupland S, John R (2010) A novel α-cut representation for type-2 fuzzy sets in fuzzy systems. In: IEEE International Conference, Barcelona, Spain\nHamrawi H (2011) Type-2 fuzzy α-cuts. Ph. D Thesis. De Montfort University\nKlir G J, Yuan B (1995) Fuzzy sets and fuzzy logic theory and applications. New Jersey: Prentice Hall PTR\nBaudrit C, Couso I, Dubois D (2007) Joint propagation of probability and possibility in risk analysis: Towards a formal framework. International Journal of Approximate Reasoning 45: 82–105\nDempster A P (1967) Upper and lower probabilities induced by a multivalued mapping. Annals of Mathematical Statistics 38: 325–339\nShafer G (1967) A mathematical theory of evidence. Princeton, NJ: Princeton University Press\nSmets P (1997) The normative representation of quantified beliefs by belief functions. Artificial Intelligence 92: 229–242\nMendel J M (2003) Type-2 fuzzy sets: Some questions and answers. IEEE Neural Networks Society: 10–13\nUSEPA (2001) Risk assessment guidance for superfund (RAGS), Vol. III: Part A. Process for conducting probabilistic risk assessment. Washington, DC: Office of Emergency and Remedial Response, EPA 540-R-02-002\nIRIS (1995) Integrated risk information system: USEPA",{"VOID":1597},"10.1007\u002Fs12543-013-0150-4","https:\u002F\u002Fwww.tandfonline.com\u002Fdoi\u002Ffull\u002F10.1007\u002Fs12543-013-0150-4",[1600],{"id":1601,"sortIndex":21,"researcher":20,"roles":1602,"affiliations":1603,"properties":1612,"displayName":1614,"givenName":20,"familyName":20},"9fe51c74-e051-4691-a46e-fb89776d9e9a",[418],[1604],{"id":1605,"sortIndex":21,"affiliation":1606,"properties":20},"16c6b198-5989-48c9-a137-7d0fd952b3f5",{"id":1605,"createTime":20,"updateTime":20,"relativeEntities":1607,"slug":20,"properties":1608,"entityType":20,"verifyStatus":20,"verifyTime":20,"verifyNote":20,"languages":20,"translateLanguages":20,"viewCount":20,"url":20,"parentIds":1611,"statistic":20},[],{"title":1609},{"VI":1610},"Department of Mathematics, Gauhati University, Guwahati, India",[],{"title":1613},{"VI":1614},"Rituparna 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