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Trapezoidal intuitionistic fuzzy prioritized aggregation operators and application to multiattribute decision making
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In some multiattribute decision making (MADM) problems, various relationships among the decision attributes should be considered. This paper investigates the prioritization relationship of attributes in MADM with trapezoidal intuitionistic fuzzy numbers (TrIFNs). TrIFNs are a special intuitionistic fuzzy set on a real number set and have the better capability to model illknown quantities. Firstly, the weighted possibility means of membership and nonmembership functions for TrIFNs are defined. Hereby, a new lexicographic ranking method for TrIFNs is presented. Then, a series of trapezoidal intuitionistic fuzzy prioritized aggregation operators are developed, including the trapezoidal intuitionistic fuzzy prioritized score (TrIFPS) operator, trapezoidal intuitionistic fuzzy prioritized weighted average (TrIFPWA) operator, trapezoidal intuitionistic fuzzy prioritized “and” (TrIFPAND) operator and trapezoidal intuitionistic fuzzy prioritized “or” (TrIFPOR) operator. Some desirable properties of these operators are also discussed. By utilizing the TrIFPWA operator, the attribute values of alternatives are integrated into the overall ones, which are used to rank the alternatives. Thus, a new method is proposed for solving the prioritized MADM problems with TrIFNs. Finally, the applicability of the proposed method is illustrated with a supply chain collaboration example.
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Shuping
Wan
Jiangxi University of Finance and Economics
Jiangxi University of Finance and Economics
China
shupingwan@163.com


Jiuying
Dong
Jiangxi University of Finance and Economics
Jiangxi University of Finance and Economics
China
jiuyingdong@126.com


Deyan
Yang
Qingdao Technological University
Qingdao Technological University
China
270632966@qq.com
Multiattribute decision making
Trapezoidal intuitionistic fuzzy number
Prioritized aggregation operators
Weighted possibility mean
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Zhang, A ranking method of triangular intuitionistic fuzzy##numbers and application to decision making, International Journal of Computational In##telligence Systems, 3(5) (2010), 522530.##[8] B. Q. Li and W. He, Intuitionistic fuzzy PRIAND and PRIOR aggregation operators,##Information Fusion, 14 (2013), 450459.##[9] R. Mansini, M. W. P. Savelsbergh and B. Tocchella, The supplier selection problem with##quantity discounts and truckload shipping, Omega, 40(4) (2012), 445455.##[10] 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.##[11] 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.##[12] S. P. Wan, D. F. Li and Z. F. Rui, Possibility mean, variance and covariance of triangular##intuitionistic fuzzy numbers, Journal of Intelligent and Fuzzy Systems, 24(4) (2013), 847##[13] S. P. Wan and D. F. Li, Possibility mean and variance based method for multiattribute##decision making with triangular intuitionistic fuzzy numbers, Journal of Intelligent and##Fuzzy Systems, 24 (2013), 743754.##[14] S. P. Wan, Q. Y. Wang and J. Y. Dong, The extended VIKOR method for multiattribute##group decision making with triangular intuitionistic fuzzy numbers, KnowledgeBased Sys##tems, 52 (2013), 6577.##[15] S. P. Wan and J. Y. Dong, Possibility method for triangular intuitionistic fuzzy multi##attribute group decision making with incomplete weight information, International Journal##of Computational Intelligence Systems, 7(1) (2014), 6579.##[16] S. P. Wan and D. F. Li. Fuzzy LINMAP approach to heterogeneous MADM considering##comparisons of alternatives with hesitation degrees, Omega, 41(6)(2013), 925940.##[17] S. P. Wan and D. F. Li. Atanassovs intuitionistic fuzzy programming method for hetero##geneous multiattribute group decision making with Atanassovs intuitionistic fuzzy truth##degrees, IEEE Transaction on Fuzzy Systems, 22(2)(2014), 300312.##[18] S. P. Wan and J. Y. Dong, Intervalvalued intuitionistic fuzzy mathematical programming##method for hybrid multicriteria group decision making with intervalvalued intuitionistic##fuzzy truth degree, Information Fusion, 26 (2015), 4965. ##[19] S. P. Wan and J. Y. Dong, A possibility degree method for intervalvalued intuitionistic##fuzzy multiattribute group decision making, Journal of Computer and System Sciences,##80(1) (2014), 237256.##[20] S. P. Wan and J. Y. Dong, Method of trapezoidal intuitionistic fuzzy number for multi##attribute group decision, Control and Decision, 25(5) (2010), 773776.##[21] S. P. Wan, Power average operators of trapezoidal intuitionistic fuzzy numbers and appli##cation to multiattribute group decision making, Applied Mathematical Modelling, 37(6)##(2013), 41124126.##[22] S. P. Wan and J. Y. Dong, Power geometric operators of trapezoidal intuitionistic fuzzy##numbers and application to multiattribute group decision making, Applied Soft Comput##ing, 29 (2015), 153168.##[23] S. P.Wan, Method based on fractional programming for intervalvalued intuitionistic trape##zoidal fuzzy number multiattribute decision making, Control and Decision, 27 (3) (2012),##[24] S. P. Wan, Multiattribute decision making method based on intervalvalued trapezoidal##intuitionistic fuzzy number, Control and Decision, 6 (2011), 857866.##[25] J. Q. Wang, Overview on fuzzy multicriteria decisionmaking approach, Control and De##cision, 23(6) (2008), 601607.##[26] J. Q. Wang and Z. Zhang, Aggregation operators on intuitionistic trapezoidal fuzzy num##ber and its application to multicriteria decision making problems, Journal of Systems##Engineering and Electronics, 20(2) (2009), 321326.##[27] J. Q.Wang, R. R. Nie, H. Y. Zhang and X. H. Chen, New operators on triangular intuition##istic fuzzy numbers and their applications in system fault analysis, Information Sciences,##251 (2013), 7995.##[28] H. M.Wang, Y. J. Xu and J. M. Merig, Prioritized aggregation for nonhomogeneous group##decision making in water resource management, Economic Computation and Economic##Cybernetics Studies and Research, 48(1) (2014), 247258.##[29] G. W. Wei, Some arithmetic aggregation operators with trapezoidal intuitionistic fuzzy##numbers and their application to group decision making, Journal of Computers, 3(2010),##[30] G. W. Wei, Hesitant fuzzy prioritized operators and their application to multiple attribute##decision making, KnowledgeBased Systems, 31(7) (2012), 176182.##[31] G. W. Wei and J. M. Merig, Methods for strategic decision making problems with immedi##ate probabilities in intuitionistic fuzzy setting, Scientia Iranica, 19(6) (2012), 19361946.##[32] J. Wu and Q. W. Cao, Some families of geometric aggregation operators with intuitionistic##trapezoidal fuzzy numbers, Applied mathematical modeling, 37 (1/2) (2013), 318327.##[33] J. Wu and Y. J. Liu, An approach for multiple attribute group decision making problems##with intervalvalued intuitionistic trapezoidal fuzzy numbers, Computers and Industrial##Engineering, 66 (2013), 311324.##[34] Y. J. Xu, T. Sun and D. F. Li, Intuitionistic fuzzy Prioritized OWA aggregation operator##and its application to multicriteria decision making, Control and Decision, 26(1) (2011),##[35] R. R. Yager, Modeling prioritized multicriteria decision making, IEEE Transactions on##Systems, Man and Cybernetics, Part B. Cybernetics, 34 (2004), 23962404.##[36] R. R. Yager, Prioritized aggregation operators, International Journal of Approximate Rea##soning, 48 (2008), 263274.##[37] R. R. Yager, Prioritized OWA aggregation, Fuzzy Optimization and Decision Making, 8##(2009), 245262.##[38] H. B. Yan, V. N. Huynh, Y. Nakamori and T. Murai, On prioritized weighted aggregation##in multicriteria decision making, Expert Systems with Applications, 38 (2011), 812823.##[39] X. H. Yu and Z. S. Xu, Prioritized intuitionistic fuzzy aggregation operators, Information##Fusion, 14(1) (2013), 108116.##[40] D. Yu, J. M. Merig and L. G. Zhou, Intervalvalued multiplicative intuitionistic fuzzy##preference relations, International Journal of Fuzzy Systems, 15(4) (2013), 412422. ##[41] D. J. Yu, Intuitionistic fuzzy prioritized operators and their application in multicriteria##group decision making, Technological and Economic Development of Economy, 19(1)##(2013), 121.##[42] D. J. Yu, Y. Y.Wu and T. Lu, Intervalvalued intuitionistic fuzzy prioritized operators and##their application in group decision making, KnowledgeBased Systems, 30 (2012), 5766.##[43] L. A. Zadeh, Fuzzy sets., Information and Control, 18 (1965), 338353.##[44] X. Zhang, F. Jian and P. D. Liu, A grey relational projection method for multiattribute##decision making based on intuitionistic trapezoidal fuzzy number, Applied Mathematical##Modelling, 37(5) (2013), 34673477.##]
An Optimization Model for Multiobjective Closedloop Supply Chain Network under uncertainty: A Hybrid Fuzzystochastic Programming Method
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In this research, we address the application of uncertaintyprogramming to design a multisite, multiproduct, multiperiod,closedloop supply chain (CLSC) network. In order to make theresults of this article more realistic, a CLSC for a case study inthe iron and steel industry has been explored. The presentedsupply chain covers three objective functions: maximization ofprofit, minimization of new products' delivery time, collectiontime and disposal time of used products, and maximizingflexibility. To solve the proposed model, an interactive hybridsolution methodology is adopted through combining a hybridfuzzystochastic programming method and a fuzzy multiobjectiveapproach. Finally, the numerical experiments are given todemonstrate the significance of the proposed model and thesolution approach.
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Behnam
Vahdani
Faculty of Industrial and Mechanical Engineering, Qazvin Branch,
Islamic Azad University, Qazvin, Iran
Faculty of Industrial and Mechanical Engineering,
Iran
b.vahdani@ut.ac.ir
Closedloop supply chain network design
Multiobjective decision making
Fuzzy mathematical programming
Stochastic programming
[[1] F. Altiparmak, M. Gen, L. Lin and T. A. Paksoy,genetic algorithm approach for multi##objective optimization of supply chain networks, Computers and Industrial Engineering, 51##(2006), 197216.##[2] B. Bilgen,Application of fuzzy mathematical programming approach to the production alloca##tion and distribution supply chain network problem, Expert Syst Appl, 37 (2010), 44884495.##[3] J. R. Birge and F. V. Louveaux,A multi cut algorithm for twostage stochastic linear pro##grams, European Journal of Operational Research, 34 (1988), 384392. ##[4] SL. Chung , HM.Wee and PC. Yang ,Optimal policy for a closedloop supply chain inventory##system with remanufacturing, Math Comput Model, 48 (2008), 867881.##[5] F. Du and GW. Evans, A biobjective reverse logistics network analysis for postsale service,##Computers and Operations Research, 35 (2008), 2634.##[6] D. Dubois and H. Prade, Possibility Theory  An Approach to the Computerized Processing##of Uncertainty, Plenum Press, New York, (1988), 2454.##[7] M. ElSayed, N. Aa and A. ElKharbotly, A stochastic model for forwardreverse logistics##network design under risk, Comput Ind Eng, 58 (2010), 423431.##[8] M. Fleischmann, P. Beullens, JM. Bloemhof ruwaard and L. Wassenhove, The impact of##product recovery on logistics network design, Production and Operations Management, 10##(1) (2001), 5673.##[9] R. A.Freeze, J. Massmann, L. Smith, J. Sperling and B. James, Hydrogeological decision##analysis, 1, a framework. Ground Water, 28 (1990), 738766.##[10] P. Guo, G. H.Huang and Y. P. Li, An inexact fuzzychanceconstrained twostage mixed##integer linear programming approach for ##ood diversion planning under multiple uncertain##ties, Advances in Water Resources, 33 (2010), 8191.##[11] G. H. Huang and D. P. Loucks, An inexact two stage stochastic programming model for water##resources management under uncertainty, Civil Engineering and Environmental Systems, 17##(2000), 95118.##[12] G. H. Huang, A hybrid inexactstochastic water management model, European Journal of##Operational Research, 107 (1998), 137158.##[13] A. Hasani, SH. Zegordi and E. Nikbakhsh, Robust closedloop supply chain network design##for perishable goods in agile manufacturing under uncertainty, Int J Prod Res, 50 (2012),##46494669.##[14] MA. Ilgin and SM. Gupta, Environmentally conscious manufacturing and product recovery##(ECMPRO): a review of the state of the art, J Environ Manage, 91 (2010), 563591.##[15] M. Inuiguchi and T. Tanino, Portfolio selection under independent possibilistic information,##Fuzzy Sets and Systems, 115 (2000), 8392.##[16] M. G. Iskander, A suggested approach for possibility and necessity dominance indices in##stochastic fuzzy linear programming, Applied Mathematics Letters, 18 (2005), 395399.##[17] S. Kara and S. A. Onut, Stochastic optimization approach for paper recycling reverse logistics##network design under uncertainty, Int J Environ Sci Technol, 4 (2010), 717730.##[18] D. Lee and M. A. Ong, Heuristic approach to logistics network design for end of lease com##puter products recovery, Transportation Research Part E, 44 (2008), 455474.##[19] DH. Lee, M. Dong and W. Bian, The design of sustainable logistics network under uncer##tainty, Int J Prod Econ, 128 (2010), 159166.##[20] W. Li, Y. P. Li, C. H. Li and G. H. Huang, An inexact twostage water management model##for planning agricultural irrigation under uncertainty, Agricultural Water Management, 97##(2010), 19051914.##[21] Y. P. Li, J. Liu and G. H. Huang,A hybrid fuzzystochastic programming method for water##trading within an agricultural system, Agricultural Systems, 123 (2014), 7183.##[22] Y. P. Li, G. H. Huang and S. L.Nie, Planning water resources management systems using##a fuzzyboundary interval stochastic programming method, Advances in Water Resources, 33##(2010), 11051117.##[23] Y. P. Li, G. H. Huang, Y. F. Huang and H. D. Zhou, A multistage fuzzystochastic program##ming model for supporting sustainable water resources allocation and management, Environ##mental Modelling and Software, 7 (2009), 786797.##[24] Y. J. Lai and C. L. Hwang, Possibilistic linear programming for managing interest rate risk,##Fuzzy Sets and Systems, 54 (1993), 135146.##[25] E. Melachrinoudis, A. Messac and H. Min, Consolidating a warehouse network: a physical##programming approach, International Journal of Production Economics, 97 (2005), 117.##[26] G. A. Mendoza, B. Bruce Bare and Z. H. Zhou, A fuzzy multiple objective linear programming##approach to forest planning under uncertainty, Agricultural Systems, 41 (1993), 257274. ##[27] EU. Olugu and KY. Wong, An expert fuzzy rulebased system for closedloop supply chain##performance assessment in the automotive industry, Expert Syst Appl, 39 (2012), 375384.##[28] S. Pokharel and A. Mutha, Perspectives on reverse logistics: a review, Resources, Conserva##tion and Recycling, 53(4) (2009) 17582.##[29] MS. Pishvaee and SA. Torabi, A possibilistic programming approach for closedloop supply##chain network design under uncertainty, Fuzzy Set Syst, 161 (2010), 26682683.##[30] MS.Pishvaee and J.Razmi, Environmental supply chain network design using multi objective##fuzzy mathematical programming, Appl Math Model, 36 (2012), 34333446.##[31] Q. Qiang, K. Ke and T. Anderson , J.Dong,The closed loop supply chain network with com##petition, distribution channel investment, and uncertainties, Omega, 41 (2013), 186194.##[32] S. Rubio, A. Chamorro and FJ. Miranda, Characteristics of the research on reverse logistics,##International Journal of Production Research, 46(4) (2008), 10991120.##[33] D. Stindt and R. Sahamie, Review of research on closed loop supply chain management in##the process industry, Flexible Services and Manufacturing Journal, 43(2) (2014), 2345.##[34] SK. Srivastava, Green supply chain management: a state of the art literature review, Inter##national Journal of Management Reviews, 9(1) (2007), 5380.##[35] MIG. Salema, AP.BarbosaPovoa and AQ.Novais, An optimization model for the design of##a capacitated multiproduct reverse logistics network with uncertainty, Eur J Oper Res, 179##(2007), 10631077.##[36] K. Subulan, AS. Tasan and A.Baykasoglu, Fuzzy mixed integer programming model for##medium term planning in a closedloop supply chain with remanufacturing option, J Intel##Fuzzy Syst, 23 (2012), 345368.##[37] S. A. Torabi and E. Hassini, An interactive possibilistic programming approach for multiple##objective supply chain master planning, Fuzzy Sets and Systems, 159 (2008), 193214.##[38] S. Verstrepen, F. Cruijssen, M. De Brito and W.Dullaert, An exploratory analysis of reverse##logistics in Flanders., European Journal of Transport and Infrastructure Research, 7(4)##(2007), 301316.##[39] B. Vahdani, J. Razmi and R. Tavakkoli Moghaddam, Fuzzy possibilistic modeling for closed##loop recycling collection networks, Environ Model Assess, 17 (2012), 623637.##[40] B. Vahdani, R. Tavakkoli Moghaddam, F. Jolai and A. Baboli, Reliable design of a closed loop##supply chain network under uncertainty: an interval fuzzy possibilistic chance constrained##model, Eng Optim, 45 (2013), 745765.##[41] B. Vahdani, R. Tavakkoli Moghaddam, M. Modarres and A. Baboli, Reliable design of a##forward/reverse logistics network under uncertainty: a robustM M c queuing model, Transp##Res Part E, 48 (2012), 11521168.##[42] P. Wells and M. Seitz, Business models and closed loop supply chains: a typology, Supply##Chain Management: An International Journal, 10(4) (2005), 249251.##[43] HF. Wang and HW. Hsu, Resolution of an uncertain closedloop logistics model: an applica##tion to fuzzy linear programs with risk analysis, J Environ Manage, 91(21) (2010), 4862.##[44] L. A. Zadeh, The concept of a linguistic variables and its application to approximate##reasoning1, Information Sciences, 8 (1975), 199249.##[45] H. J. Zimmermann, Fuzzy Set Theory and its Applications, third ed, Kluwer Academic Pub##lishers, (1996), 3247.##[46] H. J. Zimmermann, Fuzzy programming and linear programming with several objective func##tions, Fuzzy Sets and Systems, 1 (1978), 4555.##]
Developing new methods to monitor phase II fuzzy linear profiles
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In some quality control applications, the quality of a process or a product is described by the relationship between a response variable and one or more explanatory variables, called a profile. Moreover, in most practical applications, the qualitative characteristic of a product/service is vague, uncertain and linguistic and cannot be precisely stated. The purpose of this paper is to propose a method for monitoring simple linear profiles with a fuzzy and ambiguous response. To this end, fuzzy EWMA and fuzzy Hotelling's $T^2$ statistics are developed using the extension principle. To monitor phase II of fuzzy linear profiles, two methods using fuzzy hypothesis testing, are presented based on these statistics. A case study in ceramic and tile industry, is provided. A simulation study to evaluate the performance of the proposed methods in terms of average run length (ARL) criterion showed that the proposed methods are very efficient in detecting various sized shifts in process profiles.
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G.
Moghadam
Department of Industrial Engineering, Isfahan University Of Technology, Isfahan, Iran
Department of Industrial Engineering, Isfahan
Iran
g.moghadam@in.iut.ac.ir


G. A.
Raissi Ardali
Department of Industrial Engineering, Isfahan University Of
Technology, Isfahan, Iran
Department of Industrial Engineering, Isfahan
Iran
raissi@cc.iut.ac.ir


V.
Amirzadeh
Department of Statistics, Shahid Bahonar University Of Kerman, Kerman, Iran
Department of Statistics, Shahid Bahonar
Iran
v_amirzadeh@uk.ac.ir
Fuzzy qualitative profiles
Fuzzy EWMA statistic
Fuzzy Hotelling's $T^2$ statistic
Fuzzy hypothesis testing
ARL criterion
[[1] C. Croarkin and R.Varner, Measurement Assurance for Dimensional Measurements on##IntegratedCircuit Photo masks, NBS Technical Note 1164, U.S. Department of Commerce,##Washington D.C., USA, 1982.##[2] M. H. FazelZarandi and A. Alaeddini, Using Adaptive NeroFuzzy Systems to Monitor Linear##Quality Proles, J. Uncertain System, 4(2) (2010 ), 147160.##[3] Sh. Ghobadi, K. Noghondarian, R. Noorossana and S. M. Sadegh Mirhosseini, Developing##a multivariate approach to monitor fuzzy quality proles, Quality & Quantity, 48 (2014),##[4] H. Hassanpur, H. R. Maleki and M. A. Yaghoobi, A goal programming approach for fuzzy##linear regression with nonfuzzy input and fuzzy output data, Asia Pacic J. Operational##Research., 26(5) (2009), 118.##[5] S. Z. Hosseinifard, M. Abdollahian and P. Zeephongsekul, Application of articial neural##networks in linear prole monitoring, Expert Systems with Applications, 38 (2011), 4920##[6] L. Kang and S. L. Albin, Online monitoring when the process yields a linear prole, J.##Quality Technology, 32(4) (2000), 418426.##[7] K. Kim, M. A. Mahmoud and W. H. Woodall, On the monitoring of linear proles, J. Quality##Technology, 35 (2003), 317328.##[8] R. Korner and W. Nather, Linear regression with random fuzzy variables: extended classical##estimates, best linear estimates, least squares estimates, J. Information Sciences, 109 (1998),##[9] Z. Li and Z. Wang, An exponentially weighted moving average scheme with variable sampling##intervals for monitoring linear proles, Computers & Industrial Engineering, 59 (2010), 630##[10] V. Monov, B. Sokolov and S. Stefan, Grinding in Ball Mill: Modeling and Process Control,##Cybernetics and Information Technologies, 12(2) (2012).##[11] D. C. Montgomery, Introduction to Statistical Quality Control, John Wiley and Sons, New##York, 2009.##[12] S. T. A. Niaki, B. Abbasi and J. Arkat, A generalized linear statistical model approach to##monitor proles, Int. J. Engineering, Transactions A: Basics, 20(3) (2007), 233242.##[13] K. Noghondarian and Sh. Ghobadi, Developing a univariate approach to phaseI monitoring##of fuzzy quality proles, Int. J. Industrial Engineering Computations, 3 (2012), 829842.##[14] R. Noorossana, A. Saghaei and A. H. Amiri, Statistical Analysis of Prole Monitoring, John##Wiley and Sons, Inc. Hoboken, New Jersey, 2011.##[15] A. Saghaei, M. Mehrjoo and A. Amiri, ACUSUMbased method for monitoring simple linear##proles, Int. J. Advanced Manufacturing Technology, 45(11) (2009), 12521260.##[16] S. Senturk, N. Erginel, I. Kaya and C. Kahraman, Fuzzy exponentially weighted moving av##erage control chart for univariate data with a real case application, Applied Soft Computing,##22 (2014), 110##[17] S. M. Taheri and M. Are, Testing fuzzy hypotheses based on fuzzy test statistic, Soft Computing##, 13 (2009), 617625.##[18] R. Viertl, Statistical Methods for Fuzzy Data, John Wiley and Sons, Austria, 2011.##[19] J. Zhang, Z. Li and Z. Wang, Control chart based on likelihood ratio for monitoring linear##proles, Computational Statistics and Data Analysis, 53 (2009), 14401448.##[20] J. Zhu and D. K. J. Lin, Monitoring the slops of linear proles, Quality Engineering, 22(1)##(2010), 112.##[21] C. Zou, Y. Zhang and Z. Wang, Control chart based on changepoint model for monitoring##linear proles, IIE Transactions, 38(12) (2006), 10931103.##[22] C. Zou, C. Zhou, Z. Wang and F. Tsung, A selfstarting control chart for linear proles, J.##Quality Technology, 39(4) (2007), 364375.##]
Effects of Project Uncertainties on Nonlinear TimeCost Tradeoff Profile
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2
This study presents the effects of project uncertainties on nonlinear timecost tradeoff (TCT) profile of real life engineering projects by the fusion of fuzzy logic and artificial neural network (ANN) models with hybrid metaheuristic (HMH) technique, abridged as FuzzyANNHMH. Nonlinear timecost relationship of project activities is dealt with ANN models. ANN models are then integrated with HMH technique to search for Paretooptimal nonlinear TCT profile. HMH technique incorporates simulated annealing in the selection process of multiobjective genetic algorithm. Moreover, in real life engineering projects, uncertainties like management experience, labor skills, and weather conditions are commonly involved, which affect the duration and cost of the project activities. FuzzyANNHMH analyses responsiveness of nonlinear TCT profile with respect to these uncertainties. A comparison of FuzzyANNHMH is made with another method in literature to solve nonlinear TCT problem and the superiority of FuzzyANNHMH is demonstrated by results. The study gives project planners to carry out the best plan that optimizes time and cost to complete a project under uncertain environment.
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100


Bhupendra Kumar
Pathak
Department of Mathematics, Jaypee University, Anoop
shahr, Bulandshahr, 203390 India
Department of Mathematics, Jaypee University,
India
pathak.maths@gmail.com


Sanjay
Srivastava
Department of Mechanical Engineering, Dayalbagh Educational
Institute (Deemed University), Dayalbagh, Agra, 282005 India
Department of Mechanical Engineering, Dayalbagh
India
ssrivastava.engg@gmail.com
fuzzy logic
Artificial neural network
Hybrid MetaHeuristic
TimeCost Tradeoff
[[1] M. V. Arias and C. A. C. Coello, Asymptotic convergence of metaheurisitcs for multiobjective##optimization problems, Soft Comput., 10 (2005), 1001{1005.##[2] S. P. Chen, M. J. Tsai, Timecost tradeo analysis of project networks in fuzzy environments,##Eur. J. Oper.Res., 212(2) (2011), 386{397.##[3] K. Deb, Multiobjective optimization using evolutionary algorithms, NewYork: JohnWiley##and Sons, 2001.##[4] E. Eshtehardian, A. Afshar and R. Abbasnia, Fuzzybased MOGA approach to stochastic##time cost tradeo problem, Automation in Construction, 18 (2009), 692{701.##[5] L. Fausett, Fundamentals of neural networks," NJ: Prentice Hall Englewood Clis, 1994.##[6] C. W. Feng, L. Liu, and S. A. Burns, Using genetic algorithms to solve construction timecost##tradeo problems, J. Comput. Civil Eng., 11 (1997), 184{189.##[7] H. Ke, W. Ma, X. Gao and W. Xu, New fuzzy models for timecost tradeo problem, Fuzzy##Optim Decis Making, 9 (2010), 219{231.##[8] S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi, Optimization by simulated annealing, Sci.,##220 (1983), 671{680.##[9] S. S. Leu, A. T. Chen and C. H. Yang, A GAbased fuzzy optimal model for construction##timecost tradeo, Int. J. Project Manage., 19 (2001), 47{58.##[10] L. Liu, S. Burns and C. Feng, Construction timecost tradeo analysis using LP/IP hybrid##method, J. Const. Eng. Manage., 121(4) (1995), 446{454.##[11] E. H. Mamdani, Application of fuzzy logic to approximate reasoning using linguistic synthesis,##Trans. Comput., 26(12) (1977), 1182{1191. ##[12] E. H. Mamdani, Applications of fuzzy algorithms for control of simple dynamic plant, In:##Proc. IEEE, 121(12) (1974), 1585{1588.##[13] B. K. Pathak and S. Srivastava, Integrated FuzzyHMH for project uncertainties in timecost##tradeo problem, Applied Soft Computing, 21 (2014), 320329.##[14] B. K. Pathak and S. Srivastava, MOGA based timecost tradeos: responsiveness for project##uncertainties, In: Proc. CEC, (2007), 3085{3092.##[15] B. K. Pathak, S. Srivastava and K. Srivastava, Neural network embedded multiobjective ge##netic algorithm to solve nonlinear timecost tradeo problems of project scheduling, J. Sci.##Ind. Res., 67(2) (2008), 124{131.##[16] S. Srivastava, B. Pathak and K. Srivastava, Project scheduling: timecost tradeo problems,##In: Comput. Intell.Optimization, Y. Tenne and CK Goh Ed., SpringerVerlag Berlin Heidel##berg, 7 (2010), 325{357.##[17] S. Srivastava, K. Srivastava, R. S. Sharma and K. H. Raj, Modelling of hot closed die forging##of an automotive piston with ANN for intelligent manufacturing, J. Sci. Ind. Res., 63 (2004),##[18] M. O. Suliman, V. S. S. Kumar and W. Abdulal, Optimization of uncertain construction time##cost trade o problem using simulated annealing algorithm, World Congr. Inform. Commun.##Tech., (2011), 489{494.##[19] L. A. Zadeh, Fuzzy sets, Inform. and Control, 8(3) (1965), 333{353.##[20] L. A. Zadeh, Outline of a New Approach to the Analysis of a Complex System and Decision##Processes, IEEE Trans. Syst. Man Cybern., 3 (1973), 28{44.##]
On the compactness property of extensions of firstorder G"{o}del logic
2
2
We study three kinds of compactness in some variants of G"{o}del logic: compactness,entailment compactness, and approximate entailment compactness.For countable firstorder underlying language we use the Henkinconstruction to prove the compactness property of extensions offirstorder g logic enriched by nullary connective or the Baaz'sprojection connective. In the case of uncountable firstorder languagewe use the ultraproduct method to derive the compactness theorem.
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101
121


Seyed Mohammad Amin
Khatami
Department of Mathematics and Computer Science,
Amirkabir University of Technology, Tehran, Iran
Department of Mathematics and Computer Science,
Am
Iran
khatami@ipm.ir


Massoud
Pourmahdian
Department of Mathematics and Computer Science, Amirk
abir University of Technology, Tehran, Iran
Department of Mathematics and Computer Science,
Iran
pourmahd@ipm.ir
G"{o}del logic
Compactness theorem
[1] M. Baaz and R. Zach, Compact propositional Godel logics, MultipleValued Logic, 28th IEEE##International Symposium on, (1998), 108113.##[2] I. BenYaacov and A. Usvyatsov, Continuous rst order logic and local stability, Transactions##of the American Mathematical Society, 362(10) (2010), 52135259.##[3] R. Cignoli, F. Esteva and L. Godo, On Lukasiewicz logic with truth constants, Theoretical##Advances and Applications of Fuzzy Logic and Soft Computing, Springer, (2007), 869875.##[4] P. Cintula, Two notions of compactness in Godel logics, Studia Logica, 81(1) (2005), 99123.##[5] P. Cintula and M. Navara, Compactness of fuzzy logics, Fuzzy Sets and Systems, 143(1)##(2004), 5973.##[6] F. Esteva, J. Gispert, L. Godo and C. Noguera, Adding truthconstants to logics of continuous##tnorms: Axiomatization and completeness results, Fuzzy Sets and Systems, 158(6) (2007),##[7] F. Esteva, L. Godo and C. Noguera. Firstorder tnorm based fuzzy logics with truthconstants:##Distinguished semantics and completeness properties, Annals of Pure and Applied##Logic, 161(2) (2009), 185202.##[8] G. Gerla, Abstract fuzzy logic, Fuzzy Logic, Springer (2001), 1944.##[9] P. Hajek, Metamathematics of Fuzzy Logic, Kluwer Academic Pub, (1998).##[10] S. M. A. Khatami, M. Pourmahdian and N. R. Tavana, From rational Godel logic to ultrametric##logic, Journal of Logic and Computation, doi: 10.1093/logcom/exu065, 2014.##[11] M. Navara and U. Bodenhofer, Compactness of fuzzy logics, Neural Network World, Citeseer,##[12] J. Pavelka, On fuzzy logic i, ii, iii, Mathematical Logic Quarterly, 25(36,712,2529)##(1979), 4552,119134,447464.##[13] M. Pourmahdian and N. R. Tavana, Compactness in rstorder Godel logics, Journal of Logic##and Computation, 23(3) (2013), 473485.##[14] N. Preining, Complete recursive axiomatizability of Godel logics, PhD thesis, Technische##Universitat Wien, 2003.##[15] N. R. Tavana, M. Pourmahdian and F. Didevar, Compactness in rstorder Lukasiewicz##logics, Journal of Logic and Computation, 20(1) (2012), 254265.##[16] S. Willard. General topology, Courier Dover Publications, 2004.##]
Remarks on completeness of latticevalued Cauchy spaces
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2
We study different completeness definitions for two categories of latticevalued Cauchy spaces and the relations between these definitions. We also show the equivalence of a socalled completion axiom and the existence of a completion.
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123
132


Gunther
Jager
School of Mechanical Engineering, University of Applied Sciences
Stralsund, 18435 Stralsund, Germany
School of Mechanical Engineering, University
Germany
g.jager@ru.ac.za, gunther.jaeger@fhstralsund.de
$L$topology
$L$Cauchy space
Completeness
Completion
[[1] H. Boustique and G. Richardson, Regularity: Latticevalued Cauchy spaces, Fuzzy Sets and##Systems, 190 (2012), 94104. ##[2] P. V. Flores, R. N. Mohapatra and G. Richardson, Latticevalued spaces: fuzzy convergence,##Fuzzy Sets and Systems, 157 (2006), 27062714.##[3] U. Hohle and A. P. Sostak, Axiomatic foundations of xedbasis fuzzy topology, In: U. Hohle,##S. E. Rodabauch (eds.), Mathematics of Fuzzy Sets. Logic, Topology and Measure Theory,##Kluwer, Boston/Dordrecht/London, 1999.##[4] G. Jager, A category of Lfuzzy convergence spaces, Quaest. Math., 24 (2001), 501517.##[5] G. Jager, Subcategories of latticevalued convergence spaces, Fuzzy Sets and Systems, 156##(2005), 124.##[6] G. Jager, Latticevalued convergence spaces and regularity, Fuzzy Sets and Systems, 159##(2008), 24882502.##[7] G. Jager, Latticevalued Cauchy spaces and completion, Quaest. Math., 33 (2010), 5374.##[8] G. Jager, Largest and smallest T2compactications of latticevalued convergence spaces,##Fuzzy Sets and Systems, 190 (2012), 3246.##[9] G. Jager, On diagonal completion of latticevalued diagonal Cauchy spaces, Fuzzy Sets and##Systems, to appear.##[10] H. H. Keller, Die LimesUniformisierbarkeit der Limesraume, Math. Ann., 176 (1968), 334##[11] D. C. Kent and G. D. Richardson, Completions of probabilistic Cauchy spaces, Math. Japonica,##48 (1998), 399407.##[12] H. Nusser, A generalization of probabilistic uniform spaces, Appl. Cat. Structures, 10 (2002),##[13] B. Pang, The category of stratied Llter spaces, Fuzzy Sets and Systems, 247 (2014),##108126. .##[14] E. E. Reed, Completions of uniform convergence spaces, Math. Ann., 194 (1971), 83108.##[15] X. F. Yang and S. G. Li, Completion of stratied (L,M)lter tower spaces, Fuzzy Sets##Systems, 210 (2013), 2238.##]
Fixed fuzzy points of generalized Geraghty type fuzzy mappings on complete metric spaces
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2
Generalized Geraghty type fuzzy mappings oncomplete metric spaces are introduced and a fixed point theorem thatgeneralizes some recent comparable results for fuzzy mappings incontemporary literature is obtained. Example is provided to show thevalidity of obtained results over comparable classical results for fuzzymappings in fixed point theory. As an application, existence of coincidencefuzzy points and common fixed fuzzy points for hybrid pair of single valuedself mapping and a fuzzy mapping is also established.
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133
146


M.
Abbas
Department of Mathematics and Applied Mathematics, University of Pre
toria, Hatfield, Pretoria, South Africa
Department of Mathematics and Applied Mathematics,
South Africa
mujahid.abbas@up.ac.za


B.
Ali
Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield 0002, Pretoria South Africa
Department of Mathematics and Applied Mathematics,
South Africa
basit.aa@gmail.com
Fixed fuzzy point
Geraghty type
Fuzzy mapping
Fuzzy set
Approximate quantity
[[1] R. P. Agarwal, M. Meehan and D. O'Regan, Fixed point theory and applications, Cambridge##University Press, 2001.##[2] A. AminiHarandi and H. Emami, A xed point theorem for contraction type maps in partially##ordered metric spaces and application to ordinary dierential equations, Nonlinear Anal., 72##(2010), 22382242.##[3] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations##integrales, Fund. Math., 3 (1922), 133{181.##[4] R. Baskaran and P. V. Subrahmanyam, A note on the solution of a class of functional##equations, Applicable Anal., 22 (1986), 235{241.##[5] R. Bellman, Methods of nonliner analysis, vol. II, 61 of Mathematics in Science and Engi##neering, Academic Press, New York, NY, USA, 1973. ##[6] D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc., 20##(1969), 458{464.##[7] D. Dukic , Z. Kadelburg and S. Radenovic, Fixed points of Geraghty type mappings in various##generalized metric spaces, Abstract Appl. Anal., Article ID 561245 (2011), 13 pages.##[8] M. Edelstein, On xed and periodic points under contractive mappings, J. London Math.##Soc., 37 (1962), 74{79.##[9] M. Edelstein, An extension of Banach contraction principle, Proc. Amer. Math. Soc., 12 (1)##(1961), 7{10.##[10] V. D. Estruch and A. Vidal, A note on xed fuzzy points for fuzzy mappings, Rend Istit.##Univ. Trieste., 32 (2001), 3945.##[11] M. Geraghty, On contractive mappings, Proc Amer Math Soc., 40 (1973), 604{608.##[12] M. E. Gordji, M. Ramezani, Y. J. Cho and S. Pirbavafa, A generalization of Geraghty's##theorem in partially ordered metric spaces and applications to ordinary dierential equations,##Fixed Point Theory Appl., 1 (74) (2012), pages 9.##[13] M. E. Gordji, H. Baghani, H. Khodaei and M. Ramezani, Geraghty's xed point theorem for##special multivalued mappings, Thai J. Math., 10 (2012), 225{231.##[14] R. H. Haghi, Sh. Rezapour and N. Shahzad, Some xed point generalizations are not real##generalizations, Nonlinear Anal., 74 (2011), 1799{1803.##[15] S. Heilpern, Fuzzy mappings and fuzzy xed point theorems, J. Math. Anal. Appl., 83 (1981),##[16] J. Jachymski, Equivalent conditions for generalized contractions on (ordered) metric spaces,##Nonlinear Analysis: Theory, Methods Appl., 3 (74), (2011), 768{774.##[17] G. Jungck, Commuting mappings and xed points, Amer. Math Monthly, 83 (1976), 261{263.##[18] B. S. Lee and S. J. Cho, A xed point theorem for contractive type fuzzy mappings, Fuzzy##Sets and Systems, 61 (1994), 309{312.##[19] S. B. Nadler, Multivalued contraction mappings, Pacic J. Math., 30 (1969), 475{488.##[20] J. J. Nieto and R. R. Lopez, Contractive mapping theorems in partially ordered sets and##applications to ordinary dierential equations, Order, 22 (3) (2005), 223{239.##[21] S. Park, Fixed points of fcontractive maps, Rocky Mountain J. Math., 8 (4) (1978), 743{##[22] E. Rakotch, A note on contractive mappings, Proc. Amer. Math. Soc., 13 (1962), 459{465.##[23] B. E. Rhoades, A comparison of various denitions of contractive mappings, Transaction.##Amer. Math. Soc., 226 (1977), 257{290.##[24] V. M. Sehgal, A xed point theorem for mappings with a contractive iterate, Proc. Amer.##Math. Soc., 23 (3) (1969), 631{634.##[25] C. S. Sen, Fixed degree for fuzzy mappings and a generalization of Ky Fan's theorem, Fuzzy##Sets and Systems, 24 (1987), 103{112.##[26] T. Suzuki, Mizoguchi and Takahashi's xed point theorem is a real generalization of Nadler's,##J. Math. Anal. Appl., 340 (2008), 752{755.##[27] D. Turkoglu and B. E. Rhoades, A xed fuzzy point for fuzzy mapping in complete metric##spaces, Math. Commun., 10 (2005), 115{121.##[28] J. S. W. Wong, Mappings of contractive type on abstract spaces, J. Math. Anal. Appl., 37##(1972), 331340.##[29] L. A. Zadeh, Fuzzy Sets, Informations and Control, 8 (1965), 103112.##[30] E. H. Zarantonello, Solving functional equations by contractive averaging, Mathematical Re##search Center, Madison, Wisconsin, Technical Summary Report No. 160, June 1960.##[31] E. Zeidler, Nonlinear functional analysis and its applications I: Fixed Point Theorems,##Springer{Verlag, Berlin, 1986.##]
Quasicontractive Mappings in Fuzzy Metric Spaces
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2
We consider the concept of fuzzy quasicontractions initiated by '{C}iri'{c} in the setting of fuzzy metric spaces and establish fixed point theorems for quasicontractive mappings and for fuzzy $mathcal{H}$contractive mappings on Mcomplete fuzzy metric spaces in the sense of George and Veeramani.The results are illustrated by a representative example.
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147
153


A.
AminiHarandi
Department of Mathematics, University of Isfahan, Isfahan, 81745
163, Iran
Department of Mathematics, University of
Iran
a.amini@sci.ui.ac.ir


D.
Mihet
West University of Timisoara, Faculty of Mathematics and Computer
Science, Bv. V. Parvan 4, 300223, Timisoara, Romania
West University of Timisoara, Faculty of
Iran
mihet@math.uvt.ro
Fuzzy metric space
Fuzzy quasicontractive mapping
Fixed point
[[1] Lj. B. Ciric, A generalization of Banach's contraction principle, Proc. Amer. Math. Soc.,##45(2) (1974), 267273.##[2] S. Chang, Y. J. Cho and S. M. Kang, Probabilistic Metric Spaces and Nonlinear Operator##Theory, Sichuan Univ. Press, 1994.##[3] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems,##64(3) (1994), 395399.##[4] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems, 27(3) (1988),##[5] V. Gregori and A. Sapena, On xed point theorems in fuzzy metric spaces, Fuzzy Sets and##Systems, 125(2) (2002), 245252.##[6] O. Hadzic and E. Pap, Fixed point theory in probabilistic metric spaces, Mathematics and its##Applications, Kluwer Academic Publishers, Dordrecht, Boston, London, 536 (2001).##[7] F. Kiany and A. AminiHarandi, Fixed points and endpoint theorems for setvalued fuzzy##contraction maps in fuzzy metric spaces, Point Theory and Applications 2011, 2011:94.##[8] E. P. Klement, R. Mesiar and E. Pap, Triangular Norms, Trends in Logics, Kluwer Academic##Publishers, Dordrecht, Boston, London, 8 (2000).##[9] I. Kramosil and J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika, 11(5)##(1975), 336344.##[10] D. Mihet, A Banach contraction theorem in fuzzy metric spaces, Fuzzy Sets and Systems,##144(3) (2004), 431439.##[11] D. Mihet, On fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets and Systems,##158(8) (2007), 915921.##[12] D. Mihet, Fuzzy contractive mappings in nonArchimedean fuzzy metric spaces, Fuzzy Sets##and Systems, 159(6) (2008), 739744.##[13] D. Mihet, A note on fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets and##Systems, 251 (2014), 8391.##[14] J. RodrguezLopez and S. Romaguera, The Hausdor fuzzy metric on compact sets, Fuzzy##Sets and Systems, 147(2) (2004), 273283.##[15] B. Schweizer and A. Sklar, Statistical metric spaces, Pacic J. Math., 10 (1960), 313334.##[16] C. Vetro, Fixed points in weak nonArchimedean fuzzy metric spaces, Fuzzy Sets and Systems,##162(1) (2011), 8490.##[17] D. Wardowski, Fuzzy contractive mappings and xed points in fuzzy metric spaces, Fuzzy##Sets and Systems, 222 (2013), 108114.##]
Persiantranslation vol. 12, no.4, August 2015
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